MATH 105, Topics in Mathematics – Lesson Four

Lesson 4: Probability

Introduction
4.1
Random Experiments and Sample Spaces
4.2
The Multiplication Rule of Counting
4.3.
Probability
4.4
Probability Spaces with Equally Likely Outcomes
4.5
The Complement of an Event
4.6
Independent Events
4.7
Odds for and against
Homework

Introduction Go to top of page

In everyday English the use of the word “probability” is not uncommon.
The “probability of occurrence of an event”
to a statistician is what “quantified chance of
occurrence of that event” is to an ordinary person.

Most people have some intuitive idea about probability:

  1. We say that the probability that the face Head will show up in a
    coin toss is 1/2. In other words, if we toss a coin many times, in
    approximately half of the times the face Head will show up.
  2. We say that, when we roll a die, probability that
    the face 5
    will show up is 1/6. In other words, if we roll the die many times,
    the face 5 will show up approximately one in six times.
  3. We have also heard of loaded dice. For a loaded die, the probability
    of a certain face to roll over is higher than that of some other face.

Let us look at the coin-toss
experiment.

4.1 Random Experiments and Sample SpacesGo to top of page

Tossing a coin or rolling a die are examples of random experiments.
Whenever we talk about probability there is a random
experiment behind it. We talk about probability in the context
of such an experiment. Let us define it more formally.

  1. Definition

    : A

    random
    experiment

    is a procedure that produces exactly one outcome
    out of many possible outcomes. All the possible outcomes are known.
    But which outcome will result when you perform the experiment is not
    known. (A random experiment is also called a statistical experiment.)

  2. Definition

    : We use the word “

    set


    to mean a collection of objects.

  3. Definition

    : Given a random experiment,
    the set of all possible outcomes is called the

    sample
    space

    . In this class, a sample space is always denoted by “S.”

Example 4.1.1. The following are examples
of some experiments and their sample spaces.

  1. Suppose your experiment is tossing a coin. Then the outcomes are
    H and T. So the sample space is

    S={H, T}

    .
    (Remark: We will use this brace notation to list the members of the
    sample space (or a set). Please try to get used to it.)

  2. Suppose your experiment is tossing a coin twice. Then the outcomes
    are HH, HT, TH, TT. So, the sample space is

    S={HH,
    HT, TH, TT}

    .

  3. Suppose your experiment is rolling a die. Then the outcomes are
    1, 2, 3, 4, 5, 6. So, the sample space is

    S={1,
    2, 3, 4, 5, 6}

    .

  4. Suppose that your experiment is rolling a die twice. Then the possible
    outcomes are (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2),
    (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3) and so on. The sample
    space is

     
     
    |
    (1,1)
    (1,2)
    (1,3)
    (1,4)
    (1,5)
    (1,6)
    |

     
     
    |
    (2,1)
    (2,2)
    (2,3)
    (2,4)
    (2,5)
    (2,6)
    |

    S
    =
    |
    (3,1)
    (3,2)
    (3,3)
    (3,4)
    (3,5)
    (3,6)
    |

    |
    (4,1)
    (4,2)
    (4,3)
    (4,4)
    (4,5)
    (4,6)
    |

     
     
    |
    (5,1)
    (5,2)
    (5,3)
    (5,4)
    (5,5)
    (5,6)
    |

     
     
    |
    (6,1)
    (6,2)
    (6,3)
    (6,4)
    (6,5)
    (6,6)
    |

  5. Suppose your experiment is to record the number of daily road accidents
    in Lawrence. Then the possible outcomes are 0, 1, 2, 3, … and
    so on. The sample space is

    S={0, 1, 2, 3, …}

    .

  6. Suppose the experiment is to determine the gender of an unborn child
    (by ultrasonic). Then the possible outcomes are Male and Female. The
    sample space is

    S={Male, Female}

    .

  7. Suppose your experiment is to determine the blood group of a patient,
    in a lab. The possible outcomes are

    S={O, A, B,
    AB}

    .

  8. Suppose your experiment is to estimate the total wheat production
    (in tons) in Kansas. Then the possible outcomes are all the positive
    numbers.

    S={any positive number} = {x: x is a positive
    number}.

  9. Suppose your experiment is to give an estimate of the annual rainfall
    (in inches) in Lawrence. Then the possible outcomes are all the positive
    numbers.

    S={any positive numbers} = {x: x is a positive
    number}.

Events

We are getting ready to talk about probability. Given a sample space,
we plan to talk about probability of an outcome. We may also talk about
the probability of EVENTS.

What is an EVENT for us? We have the following
definitions:

  1. Definition

    : Given a sample space, an

    event

    is a collection of outcomes. An event is a subcollection (or subset)
    from the sample space.

  2. Definition

    : When we perform a random experiment
    exactly one outcome results. If E is an event, then

    we
    say that E occurred if the outcome is a member of E

    .

  3. Definitions

    : There are two special events.

    1. First, there is an event (denoted by

      ø,

      called

      impossible event

      . The impossible
      event has no outcome in it. That means it is “empty.” The impossible
      event never occurs.

    2. The whole sample space S is also an event to be called

      certain
      event

      . The certain event occurs whenever you perform the
      experiment.

Examples 4.1.2.

The following are some examples
of events with reference to the examples 4.1.1 of sample spaces above.

  1. Look at the example of tossing a coin (1). Then {H} is an event,
    and so is {T}.
  2. Refer to the example of tossing a coin twice (2). Let E be the
    event that there was at least one T, and let F be the event that both
    the tosses produced the same face. Then E={HT, TH, TT} and F = {HH,
    TT}.
  3. Refer to the example of rolling a die (3). Let E={1,2,3}. Then
    E is an event. E can be described as the event that the “face value”
    was less or equal to 3.
  4. Refer to the example of rolling a die twice (4). Let E be the event
    that the first die showed the face 5. Then E={(5,1), (5,2), (5,3),
    (5,4), (5,5), (5,6)}. Let T be the event that the sum of the two “face
    values” is 5. Then T={(1,4), (2,3), (3,2), (4,1)}.
  5. Refer to the example on road accidents (5). Let E be the event
    that there was no accident in Lawrence on a day. Then E={0}.
  6. Refer to the example on annual rainfall in Lawrence. I define a
    year as a “dry year” if the annual rainfall is less than 5 inches.
    Let E be the event that a given year will be a dry year. Then E is
    the set of all positive numbers from 0 to 5.

For now, we will deal with sample spaces that have only a finite number
of outcomes. We are still getting ready to talk about probability. In
certain cases, computing probability of an event involves counting the
number of outcomes in the event and the sample space. We need to learn
a little bit about counting techniques.

4.3. Probability Go to top of page

The following are some examples of events with reference to the examples 4.1.1 of sample spaces above.For now, we will deal with sample spaces that have only a finite number of outcomes. We are still getting ready to talk about probability. In certain cases, computing probability of an event involves counting the number of outcomes in the event and the sample space. We need to learn a little bit about counting techniques.

Now we are ready to talk about probability of an outcome or an event.
If we toss a coin, then one believes that the probability that the face
Head will show up is 1/2. But this is about a “normal” coin. What if
we toss a loaded coin? If you have a loaded coin, you may
know that the probability that the face Head will show up is
1/5. But what does this mean? How and where did you learn that, for
your loaded die, the probab i l i t y t h a t t h e f a c e H e a d w i l l s h o w u p i s
1 / 5 ? W h e n a n o r d i n a r y p e r s o n m a k e s s u c h p r o b a b i l i t y s t a t e m e n t s h e / s h e
i s , i n f a c t , t a l k i n g a b o u t h i s / h e r e x p e r i e n c e . R e g a r d i n g y o u r l o a d e d
c o i n , y o u h a v e t o s s e d y o u r c o i n m a n y t i m e s a n d h a v e e x p e r i e n c e d t h a t
a b o u t o n c e i n f i v e t o s s e s t h e f a c e H e a d s h o w e d u p a n d o t h e r t i m e s t h e
f a c e T a i l s h o w e d u p . T h e r e f o r e , ” y o u k n o w ” t h a t t h e p r o b a b i l i t y t h a t
t h e f a c e H e a d w i l l s h o w u p i s 1 / 5 . / p >
p > S i m i l a r l y , y o u r o l l e d a d i e m a n y t i m e s . Y o u h a v e e x p e r i e n c e d t h a t a b o u t
o n c e i n s i x t h r o w s t h e f a c e 4 s h o w s u p . T h e r e f o r e , ” y o u k n o w ” t h a t t h e
p r o b a b i l i t y t h a t t h e f a c e 4 w i l l s h o w u p w h e n y o u t h r o w t h e d i e i s 1 / 6 .
I t i s a d i f f e r e n t s t o r y i f y o u a r e w o r k i n g w i t h a l o a d e d d i e b e c a u s e
y o u r e x p e r i e n c e w o u l d t e l l y o u s o m e t h i n g d i f f e r e n t . / p >
p > T h a t w a s t h e ” r e a l l i f e ” i d e a o f p r o b a b i l i t y . I n t h e s t u d y o f t h e m a t h e m a t i c s
o f p r o b a b i l i t y , w e a c c e p t t h i s ” e x p e r i e n c e ” a s p a r t o f t h e ” p r o b a b i l i t y
m o d e l ” a n d d o t h e m a t h e m a t i c s . / p >
p > s p a n c l a s s = ” b o l d r e d ” > T h e m a t h e m a t i c s o f p r o b a b i l i t y i n c l u d e s t h e f o l l o w i n g / s p a n > :
/ p >
o l >
l i > A d e s c r i p t i o n o f t h e s a m p l e s p a c e S ( a n d / o r t h e r a n d o m e x p e r i m e n t ) .
/ l i >
l i > A m e t h o d o r f o r m u l a t o c o m p u t e t h e p r o b a b i l i t y o f a n e v e n t . G i v e n
a n e v e n t E , t h e m e t h o d o r t h e f o r m u l a g i v e s u s t h e s y s t e m o f c o m p u t i n g
t h e p r o b a b i l i t y P ( E ) o f E . T h e p r o b a b i l i t i e s P ( E ) m u s t s a t i s f y c e r t a i n
l a w s o f p r o b a b i l i t y . / l i >
l i > A s p a n c l a s s = ” r e d ” > p r o b a b i l i t y s p a c e / s p a n > i s a s a m p l e s p a c e
S
w i t h p r o b a b i l i t y a s s i g n m e n t a s i n ( 2 ) . T o d e s c r i b e a p r o b a b i l i t y s p a c e ,
w e h a v e t o g i v e ( 1 ) a n d ( 2 ) . / l i >
l i > s p a n c l a s s = ” r e d ” > L a w s o f P r o b a b i l i t y / s p a n > : L e t b r >
c e n t e r >
s p a n c l a s s = ” b o l d r e d ” > S = { o s u b > 1 / s u b > , o s u b > 2 / s u b > , . . . , o s u b > N / s u b > } / s p a n >
/ c e n t e r >
b r >
b e a ( f i n i t e ) s a m p l e s p a c e . ( H e r e o s u b > 1 / s u b > , o s u b > 2 / s u b > , . . . ,
o s u b > N / s u b > a r e t h e o u t c o m e s o f t h e e x p e r i m e n t s . ) F o l l o w i n g a r e
t h e e l e m e n t s o f p r o b a b i l i t y s p a c e s : / l i >
l i > F o r e a c h o u t c o m e o s u b > i / s u b > , a m e t h o d o r a f o r m u l a i s g i v e n s o
t h a t w e c a n c o m p u t e t h e n u m b e r P ( o s u b > i / s u b > ) , t o b e c a l l e d t h e
p r o b a b i l i t y t h a t t h e o u t c o m e w a s o s u b > i / s u b > . T h e p r o b a b i l i t i e s
P ( o s u b > i / s u b > ) m u s t s a t i s f y t h e f o l l o w i n g t w o p r o p e r t i e s :
o l t y p e = ” a ” >
l i > s p a n c l a s s = ” b o l d ” > P ( o s u b > i / s u b > ) i s a n u m b e r b e t w e e n 0 a n d
1 / s p a n > ; / l i >
l i > s p a n c l a s s = ” b o l d ” > P ( o s u b > 1 / s u b > ) + P ( o s u b > 2 / s u b > ) + . . . +
P ( o s u b > N / s u b > ) , = 1 / s p a n > . b r >
T h e s u m o f p r o b a b i l i t i e s o f a l l t h e o u t c o m e s i s 1 .
p > / p >
/ l i >
/ o l >
/ l i >
l i > G i v e n a n e v e n t E , t h e p r o b a b i l i t y P ( E ) o f E i s d e f i n e d a s t h e s u m
o f p r o b a b i l i t i e s o f a l l t h e o u t c o m e s i n E . b r >
b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d > s p a n c l a s s = ” b o l d r e d ” > P ( E ) = t h e s u m o f p r o b a b i l i t i e s o f a l l
t h e o u t c o m e s i n E / s p a n > . / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
b r >
/ l i >
l i > T h e p r o b a b i l i t y o f t h e i m p o s s i b l e e v e n t i s z e r o , a n d t h e p r o b a b i l i t y
o f t h e c e r t a i n e v e n t i s o n e . b r >
t a b l e a l i g n = ” c e n t e r ” w i d t h = ” 4 5 0 ” >
t b o d y > t r >
t d c l a s s = ” b o l d r e d ” w i d t h = ” 1 8 0 ” a l i g n = ” l e f t ” > P ( i m p o s s i b l e e v e n t )
= 0 / t d >
t d w i d t h = ” 2 0 9 ” a l i g n = ” r i g h t ” > s p a n c l a s s = ” b o l d r e d ” > P ( c e r t a i n
e v e n t ) = P ( S ) = 1 / s p a n > . / t d >
/ t r >
/ t b o d y > / t a b l e >
/ l i >
/ o l >
p c l a s s = ” b o l d ” > 4 . 3 . P r o b l e m s o n P r o b a b i l i t y / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 3 . l . / s p a n > T h e f o l l o w i n g t a b l e g i v e s
t h e b l o o d g r o u p d i s t r i b u t i o n o f a c e r t a i n p o p u l a t i o n . b r >
/ p >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t d c o l s p a n = ” 5 ” > B l o o d G r o u p D i s t r i b u t i o n b r >
s o u r c e : a h r e f = ” h t t p : / / w w w . b o d y w a t c h . c o . u k / b l o o d g r o u p / b l o o d g r o u p 1 . h t m l ” > B o d y W a t c h / a >
/ t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t h > B l o o d G r o u p / t h >
t d > O / t d >
t d > A / t d >
t d > B / t d >
t d > A B / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t h > P e r c e n t a g e o f b r >
P o p u l a t i o n / t h >
t d > 4 7 / t d >
t d > 4 2 / t d >
t d > 8 / t d >
t d > 3 / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
S u p p o s e y o u d e t e r m i n e t h e b l o o d g r o u p o f a r a n d o m l y s e l e c t e d p e r s o n f r o m
t h i s p o p u l a t i o n .
o l >
l i > W h a t i s t h e s a m p l e s p a c e ? / l i >
l i > W h a t a r e t h e p o s s i b l e o u t c o m e s o f t h e e x p e r i m e n t a n d t h e i r p r o b a b i l i t i e s ?
/ l i >
l i > W r i t e d o w n t h e e v e n t t h a t t h e s e l e c t e d p e r s o n h a s b l o o d g r o u p A
o r B o r A B i n b r a c e n o t a t i o n . / l i >
l i > F i n d t h e p r o b a b i l i t y t h a t a r a n d o m s a m p l e o f b l o o d i s o f b l o o d
g r o u p s p a n c l a s s = ” b o l d ” > A o r B o r A B / s p a n > . / l i >
/ o l >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 . h t m l ” > S o l u t i o n / a >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 3 . 2 . / s p a n > A s t u d e n t w a n t s t o p i c k a
s c h o o l b a s e d o n t h e p a s t g r a d e d i s t r i b u t i o n o f t h e s c h o o l . F o l l o w i n g
i s a g r a d e d i s t r i b u t i o n o f l a s t y e a r i n a s c h o o l : b r >
/ p >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t d c o l s p a n = ” 6 ” > G r a d e D i s t r i b u t i o n b r >
U n r e a l D a t a / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t h > G r a d e s / t h >
t d > A / t d >
t d > B / t d >
t d > C / t d >
t d > D / t d >
t d > F / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t h > P e r c e n t a g e o f b r >
S t u d e n t s / t h >
t d > 1 9 / t d >
t d > 3 3 / t d >
t d > 3 1 / t d >
t d > 1 4 / t d >
t d > 3 / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
S u p p o s e y o u n o t e d o w n t h e g r a d e o f a r a n d o m l y s e l e c t e d s t u d e n t f r o m t h i s
s c h o o l .
o l >
l i > W h a t i s t h e s a m p l e s p a c e ? / l i >
l i > W h a t a r e t h e p o s s i b l e o u t c o m e s o f t h e e x p e r i m e n t a n d t h e i r p r o b a b i l i t i e s ?
/ l i >
l i > W r i t e d o w n t h e e v e n t t h a t t h e s e l e c t e d s t u d e n t ‘ s g r a d e i s a t l e a s t
a B . / l i >
l i > F i n d t h e p r o b a b i l i t y t h a t a r a n d o m l y p i c k e d s t u d e n t h a s a t l e a s t
a B a v e r a g e . / l i >
/ o l >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 2 . h t m l ” > S o l u t i o n / a >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 3 . 3 . / s p a n > T h e f o l l o w i n g t a b l e g i v e s
t h e p r o b a b i l i t y d i s t r i b u t i o n o f a l o a d e d d i e . b r >
/ p >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t d c o l s p a n = ” 7 ” > P r o b a b i l i t y D i s t r i b u t i o n o f a D i e / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t h > F a c e / t h >
t d > 1 / t d >
t d > 2 / t d >
t d > 3 / t d >
t d > 4 / t d >
t d > 5 / t d >
t d > 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t h > P r o b a b i l i t y / t h >
t d > 0 . 2 0 / t d >
t d > 0 . 1 5 / t d >
t d > 0 . 1 5 / t d >
t d > 0 . 1 0 / t d >
t d > 0 . 0 5 / t d >
t d > 0 . 3 5 / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
o l >
l i > W h a t i s t h e s a m p l e s p a c e ? / l i >
l i > W r i t e d o w n t h e e v e n t t h a t t h e d i e s h o w s 2 o r 3 o r 6 , i n b r a c e n o t a t i o n .
/ l i >
l i > F i n d t h e p r o b a b i l i t y t h a t t h e f a c e 2 o r 3 o r 6 s h o w s u p w h e n y o u
r o l l t h e d i e . / l i >
/ o l >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 3 . h t m l ” > S o l u t i o n / a >
! – – E N D S w i t c h P r o b a l i l i t y S e c t i o n – – >

h 3 > a n a m e = ” 2 ” > / a > 4 . 2 T h e M u l t i p l i c a t i o n R u l e o f C o u n t i n g a h r e f = ” # t o p ” > i m g s r c = ” . / i m a g e s / u p . g i f ” w i d t h = ” 2 0 ” h e i g h t = ” 1 3 ” a l t = ” G o t o t o p o f p a g e ” b o r d e r = ” 0 ” > / a > / h 3 >
p >
A s o m e p r o b a b i l i t y p r o b l e m s i n v o l v e c o u n t i n g .
S o , w e w i l l s p e n d s o m e t i m e o f d i f f e r e n t m e t h o d s o f c o u n t i n g .
/ p >

p > T h e m u l t i p l i c a t i o n r u l e s t a t e s t h a t w h e n s o m e t h i n g t a k e s p l a c e i n
s e v e r a l s t a g e s , t o f i n d s p a n c l a s s = ” r e d ” > t h e t o t a l n u m b e r o f w a y s i t
c a n o c c u r w e m u l t i p l y t h e n u m b e r o f w a y s e a c h i n d i v i d u a l s t a g e c a n o c c u r / s p a n > .
/ p >
p > I t g o e s a s f o l l o w s : S u p p o s e a c e r t a i n j o b ( o f t e n a s e l e c t i o n o f a n
i t e m ) i s a c c o m p l i s h e d i n r s t a g e s . / p >
u l >
l i > T h e f i r s t s t a g e o f t h e j o b c a n b e a c c o m p l i s h e d i n n s u b > 1 / s u b >
w a y s . b r >
/ l i >
l i > T h e s e c o n d s t a g e o f t h e j o b c a n b e a c c o m p l i s h e d i n n s u b > 2 / s u b >
w a y s . b r >
/ l i >
l i > T h e t h i r d s t a g e o f t h e j o b c a n b e a c c o m p l i s h e d i n n s u b > 3 / s u b >
w a y s . b r >
. . . . . . . . . b r >
/ l i >
l i > T h e f i n a l r – t h s t a g e o f t h e j o b c a n b e a c c o m p l i s h e d i n n s u b > r / s u b >
w a y s . / l i >
/ u l >
p > T h e n t h e n u m b e r o f w a y s t h e o r i g i n a l j o b c a n b e a c c o m p l i s h e d i s / p >
p c l a s s = ” c e n t e r ” > s p a n c l a s s = ” b o l d ” > n s u b > 1 / s u b > n s u b > 2 / s u b > n s u b > 3 / s u b >
. . . n s u b > r / s u b > w a y s . / s p a n > / p >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 2 . 1 . / s p a n > A h o u s e h o l d w o u l d l i k e t o
i n s t a l l a s t o r m d o o r . T h e l o c a l s t o r e o f f e r s t w o b r a n d n a m e s ; e a c h b r a n d
h a s f o u r d i f f e r e n t s t y l e s a n d t h r e e c o l o r s . F i n d h o w m a n y c h o i c e s h e
h a s i n t h e s e l e c t i o n . a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 1 . h t m l ” > S o l u t i o n
/ a > b r >
/ p >
p a l i g n = ” j u s t i f y ” > T h e j o b o f p i c k i n g t h e d o o r i s d o n e i n t h r e e s t a g e s .
T h e f i r s t s t a g e i s t o p i c k t h e b r a n d , w h i c h w e c a n d o i n t w o w a y s . T h e
s e c o n d s t a g e i s t o p i c k t h e s t y l e , w h i c h w e c a n d o i n f o u r w a y s . T h e n ,
t h e t h i r d s t a g e i s t o p i c k t h e c o l o r , w h i c h w e c a n d o i n t h r e e w a y s . b r >
/ p >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > P i c k t h e b r a n d / t d >
t d > 2 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > P i c k t h e s t y l e / t d >
t d > 4 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 3 / t d >
t d > P i c k t h e c o l o r / t d >
t d > 3 / t d >
/ t r >
/ t b o d y > / t a b l e >
p > T h e w h o l e j o b o f p i c k i n g t h e d o o r r c a n b e d o n e i n 2 x 4 x 3 = 2 4 w a y s . / p >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 2 . 2 . / s p a n > R e f e r t o t h e e x a m p l e ( 1 . 4 )
o f r o l l i n g a d i e t w i c e . W e w a n t t o c o u n t t h e n u m b e r o f o u t c o m e s i n t h e
s a m p l e s p a c e S . T h e w h o l e e x p e r i m e n t c o u l d b e a c c o m p l i s h e d i n t w o s t a g e s .
F i r s t , t h e d i e i s r o l l e d , a n d t h e n u m b e r o f o u t c o m e s f o r t h i s f i r s t
s t a g e i s s i x . T h e s e c o n d s t a g e i s t o r o l l t h e d i e a g a i n ; t h e s e c o n d
s t a g e a l s o h a s s i x o u t c o m e s . / p >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > R o l l t h e d i e / t d >
t d > 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > R o l l t h e d i e a g a i n / t d >
t d > 6 / t d >
/ t r >
/ t b o d y > / t a b l e >
p > T h e t o t a l n u m b e r o f o u t c o m e s i n S , b y t h e m u l t i p l i c a t i o n p r i n c i p l e ,
i s 6 x 6 = 3 6 . / p >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 2 . 3 . / s p a n > I w a n t t o a s s i g n t h e 1 0 s e a t s
o n t h e f i r s t r o w t o t h e 1 6 3 s t u d e n t s i n t h e c l a s s . H o w m a n y w a y s w e
c a n d o i t ? / p >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > A s s i g n t h e f i r s t s e a t / t d >
t d > 1 6 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > A s s i g n t h e 2 n d s e a t / t d >
t d > 1 6 2 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 3 / t d >
t d > A s s i g n t h e 3 r d s e a t / t d >
t d > 1 6 1 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 4 / t d >
t d > A s s i g n t h e 4 t h s e a t / t d >
t d > 1 6 0 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 5 / t d >
t d > A s s i g n t h e 5 t h s e a t / t d >
t d > 1 5 9 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 6 / t d >
t d > A s s i g n t h e 6 t h s e a t / t d >
t d > 1 5 8 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 7 / t d >
t d > A s s i g n t h e 7 t h s e a t / t d >
t d > 1 5 7 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 8 / t d >
t d > A s s i g n t h e 8 t h s e a t / t d >
t d > 1 5 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 9 / t d >
t d > A s s i g n t h e 9 t h s e a t / t d >
t d > 1 5 5 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 0 / t d >
t d > A s s i g n t h e 1 0 t h s e a t / t d >
t d > 1 5 4 / t d >
/ t r >
/ t b o d y > / t a b l e >
p > S o t h e t o t a l n u m b e r o f w a y s t h i s c a n b e d o n e i s = / p >
p c l a s s = ” c e n t e r ” > 1 6 3 x 1 6 2 x 1 6 1 x 1 6 0 x 1 5 9 x 1 5 8 x 1 5 7 x 1 5 6 x 1 5 5
x 1 5 4 / p >
p > s p a n c l a s s = ” b o l d ” > R e m a r k / s p a n > . T h e m u l t i p l i c a t i o n r u l e o f c o u n t i n g
h a s w i d e a p p l i c a t i o n s . Y o u m u s t c o r r e c t l y i d e n t i f y w h e t h e r y o u r c o u n t i n g
p r o b l e m c a n b e d i v i d e d i n t o s e v e r a l s t a g e s o f s i m p l e c o u n t i n g p r o b l e m s .
C o n f u s i o n m a y a r i s e a s f o l l o w s . / p >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 2 . 4 . / s p a n > S u p p o s e t h a t w e w a n t t o f o r m
a c o m m i t t e e o f 1 0 s t u d e n t s o u t o f t h e 1 6 3 s t u d e n t s i n t h i s c l a s s . W e
h a v e j u s t a s s i g n e d t h e 1 0 s e a t s i n t h e f i r s t r o w t o t h e 1 6 3 s t u d e n t s ,
w h i c h c a n b e d o n e i n 1 6 3 x 1 6 2 x . . . x 1 5 4 w a y . C o u l d w e s a y t h a t t h e
n u m b e r o f w a y s w e c a n f o r m a c o m m i t t e e o f 1 0 o u t o f t h e 1 6 3 s t u d e n t s
i n t h i s c l a s s s p a n c l a s s = ” r e d ” > i s t h e s a m e / s p a n > ? s p a n c l a s s = ” r e d ” > T h e
a n s w e r i s N O / s p a n > . W h i l e w e a s s i g n e d t h e s e a t s , t h e d i f f e r e n t a s s i g n m e n t s
o f t h e s e a t s , i n t h e f i r s t r o w , t o t h e s p a n c l a s s = ” r e d ” > s a m e g r o u p / s p a n >
o f 1 0 s t u d e n t s i s c o n s i d e r e d a s d i s t i n c t . W h i l e f o r m i n g a c o m m i t t e e ,
t h e g r o u p o f 1 0 a s a w h o l e i s c o u n t e d a s o n e c o m m i t t e e . W i t h o u t g o i n g
i n t o d e t a i l s , t h e n u m b e r o f w a y s s u c h a c o m m i t t e e c a n b e f o r m e d i s b r >
/ p >
p c l a s s = ” c e n t e r ” > s p a n c l a s s = ” b o l d ” > ( 1 6 3 x 1 6 2 x . . . x 1 5 4 ) / ( 1 x 2
x . . . x 1 0 ) . / s p a n > / p >
h 3 > O r d e r e d a n d u n o r d e r e d s e l e c t i o n / h 3 >
p > M a n y c o u n t i n g p r o b l e m s t h a t w e c o n s i d e r e s s e n t i a l l y a r e l i k e s e l e c t i n g
r o b j e c t s ( o r p e o p l e ) f r o m a c o l l e c t i o n o f n o b j e c t s ( o r p e o p l e ) . T h e r e
a r e t w o t y p e s o f s u c h s e l e c t i o n s . I n e x a m p l e ( 4 . 2 . 3 ) , t h e a s s i g n m e n t
o f t h e 1 0 s e a t s i s s e l e c t i o n o f 1 0 s t u d e n t s w h e r e t h e o r d e r i n w h i c h
w e s e l e c t e d 1 0 s t u d e n t s d i d m a t t e r . H o w e v e r , i n e x a m p l e ( 4 . 2 . 4 ) o r d e r
i n w h i c h w e p i c k 1 0 t o r e p r e s e n t i n t h e c o m m i t t e e d i d n o t c o u n t . T h e
s e l e c t i o n i n ( 4 . 2 . 3 ) i s a n s p a n c l a s s = ” r e d ” > o r d e r e d – s e l e c t i o n / s p a n >
o f r ” o b j e c t s ” f r o m a g r o u p n ” o b j e c t s . ” B u t t h e s e l e c t i o n i n ( 4 . 2 . 4 )
i s a n s p a n c l a s s = ” r e d ” > u n o r d e r e d – s e l e c t i o n / s p a n > o f r ” o b j e c t s ” f r o m
a g r o u p o f n ” o b j e c t s . ” / p >
p > s p a n c l a s s = ” b o l d ” > W e h a v e t h e f o l l o w i n g d e f i n i t i o n s , f o r m u l a s , a n d
n o t a t i o n s i n t h i s c o n t e x t / s p a n > : / p >
o l >
l i > s p a n c l a s s = ” b o l d ” > D e f i n i t i o n 1 / s p a n > . A s e l e c t i o n o f r o b j e c t s
f r o m a c o l l e c t i o n o f n o b j e c t s w h e r e d i f f e r e n t o r d e r o f s e l e c t i o n
c o u n t s a s d i s t i n c t i s c a l l e d a n s p a n c l a s s = ” r e d ” > o r d e r e d – s e l e c t i o n / s p a n > .
A n o r d e r e d s e l e c t i o n i s a l s o c a l l e d a s p a n c l a s s = ” r e d ” > p e r m u t a t i o n / s p a n > .
A n o r d e r e d – s e l e c t i o n o f r o b j e c t s f r o m a g r o u p o f n o b j e c t s i s s p a n c l a s s = ” r e d ” > c a l l e d
a p e r m u t a t i o n o f n o b j e c t s t a k e n r a t a t i m e / s p a n > . A n a s s i g n m e n t
o f t h e 1 0 s e a t s i n t h e f i r s t r o w t o 1 6 3 s t u d e n t s i s a p e r m u t a t i o n
o f 1 6 3 s t u d e n t s t a k e n 1 0 a t a t i m e . / l i >
l i > s p a n c l a s s = ” b o l d ” > D e f i n i t i o n 2 / s p a n > . A s e l e c t i o n o f r o b j e c t s
f r o m a c o l l e c t i o n o f n o b j e c t s w h e r e d i f f e r e n t o r d e r o f s e l e c t i o n
s p a n c l a s s = ” r e d ” > d o e s n o t / s p a n > c o u n t a s d i s t i n c t i s c a l l e d a n s p a n c l a s s = ” r e d ” > u n o r d e r e d – s e l e c t i o n / s p a n > .
A n u n o r d e r e d – s e l e c t i o n i s a l s o c a l l e d a s p a n c l a s s = ” r e d ” > c o m b i n a t i o n / s p a n > .
A n u n o r d e r e d – s e l e c t i o n o f r o b j e c t s f r o m a g r o u p o f n o b j e c t s i s s p a n c l a s s = ” r e d ” > c a l l e d
a c o m b i n a t i o n o f n o b j e c t s t a k e n r a t a t i m e / s p a n > . A p a r t i c u l a r
c o m m i t t e e o f 1 0 f o r m e d o u t o f 1 6 3 s t u d e n t s i s a c o m b i n a t i o n o f 1 6 3
s t u d e n t s t a k e n 1 0 a t a t i m e . / l i >
l i > s p a n c l a s s = ” b o l d ” > N o t a t i o n / s p a n > : S u p p o s e n i s a p o s i t i v e i n t e g e r .
T h e p r o d u c t o f a l l i n t e g e r s f r o m o n e t h r o u g h n i s c a l l e d ” f a c t o r i a l
n ” a n d i s d e n o t e d b y n ! . b r >
p c l a s s = ” c e n t e r ” > s p a n c l a s s = ” b o l d ” > n ! = 1 x 2 x . . . x ( n – 1 ) x
n b r >
A l s o 0 ! = 1 / s p a n > . / p >
/ l i >
l i > T h e n u m b e r o f ( p o s s i b l e ) p e r m u t a t i o n s o f n o b j e c t s t a k e n r a t a
t i m e i s d e n o t e d b y s u b > n / s u b > P s u b > r / s u b > . b r >
p c l a s s = ” c e n t e r ” > s p a n c l a s s = ” b o l d ” > s u b > n / s u b > P s u b > r / s u b > = n ! / ( n – r ) ! / s p a n >
s p a n c l a s s = ” b o l d r e d ” > &n b s p ; = &n b s p ; n x ( n – 1 ) x . . . x ( n – r + 1 )
/ s p a n > / p >
/ l i >
l i > T h e n u m b e r o f ( p o s s i b l e ) c o m b i n a t i o n s o f n o b j e c t s t a k e n r a t a
t i m e i s d e n o t e d b y s u b > n / s u b > C s u b > r / s u b > . b r >
b r >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 0 ” w i d t h = ” 5 4 3 ” >
t r a l i g n = ” c e n t e r ” >
t d h e i g h t = ” 2 9 ” > s u b > s p a n c l a s s = ” b o l d ” > n / s p a n > / s u b > s p a n c l a s s = ” b o l d ” > C s u b > r / s u b >
= s u b > n / s u b > P s u b > r / s u b > / r ! / s p a n > / t d >
t d > &n b s p ; s p a n c l a s s = ” b o l d r e d ” > = n ! / ( r ! ( n – r ) ! ) / s p a n > / t d >

t d > &n b s p ; s p a n c l a s s = ” b o l d ” > = ( n x ( n – 1 ) x . . . x ( n – r + 1 ) ) / ( 1
x 2 x . . . x r ) / s p a n > / t d >
/ t r >
/ t a b l e >

/ l i >
/ o l >
p > s p a n c l a s s = ” b o l d ” > P r o b l e m s o n 4 . 2 T h e M u l t i p l i c a t i o n R u l e o f C o u n t i n g / s p a n > / p >

B e f o r e y o u a t t e m p t a n y p r o b l e m , r e v i e w t h e a h r e f = ” . / c o u n t i n g . h t m l ” > d i a g r a m . / a >

p c l a s s = ” b o l d ” > E x e r c i s e 4 . 2 . 1 . / p >
o l t y p e = ” a ” >
l i > C o m p u t e 3 ! , 6 ! , 8 ! . / l i >
l i > C o m p u t e s u b > 5 / s u b > P s u b > 2 / s u b > , s u b > 7 / s u b > P s u b > 4 / s u b > , s u b > 6 / s u b > P s u b > 2 / s u b > ,
s u b > 4 / s u b > P s u b > 3 / s u b > . / l i >
l i > C o m p u t e s u b > 5 / s u b > C s u b > 2 / s u b > , s u b > 7 / s u b > C s u b > 4 / s u b > , s u b > 6 / s u b > C s u b > 2 / s u b > ,
s u b > 4 / s u b > C s u b > 3 / s u b > . / l i >
/ o l >
p c l a s s = ” b o l d ” > E x e r c i s e 4 . 2 . 2 . / p >
p > L e t m e c o m p u t e s u b > 8 / s u b > P s u b > 5 / s u b > : / p >
o l t y p e = ” a ” >
l i > F i r s t m e t h o d : s u b > 8 / s u b > P s u b > 5 / s u b > = 8 x &# 8 2 3 0 ; x ( 8 – 5 + 1 )
= 8 x &# 8 2 3 0 ; x 4 = 8 x 7 x 6 x 5 x 4 = 6 7 2 0 .
p > / p >
/ l i >
l i > S e c o n d M e t h o d : s u b > 8 / s u b > P s u b > 5 / s u b > = 8 ! / ( 8 – 5 ) ! = 8 ! / 3 ! = ( 1
x 2 x &# 8 2 3 0 ; x 8 ) / ( 1 x 2 x 3 ) = 4 0 3 2 0 / 6 = 6 7 2 0 / l i >
/ o l >
p c l a s s = ” b o l d ” > E x e r c i s e 4 . 2 . 3 . / p >
p > L e t m e a l s o c o m p u t e s u b > 9 / s u b > C s u b > 4 / s u b > . I l i k e t o c o m p u t e a s
f o l l o w s : s u b > 9 / s u b > C s u b > 4 / s u b > = s u b > 9 / s u b > P s u b > 4 / s u b > / 4 ! .
b r >
W e h a v e s u b > 9 / s u b > P s u b > 4 / s u b > = 9 x 8 x 7 x 6 = 3 0 2 4 b r >
a n d b r >
4 ! = 1 x 2 x 3 x 4 = 2 4 . b r >
S o , s u b > 9 / s u b > C s u b > 4 / s u b > = s u b > 9 / s u b > P s u b > 4 / s u b > / 4 ! = 3 0 2 4 / 2 4
= 1 2 6 . / p >
p c l a s s = ” b o l d ” > E x e r c i s e 4 . 2 . 4 . / p >
p > H o w m a n y w a y s c a n y o u d e a l a h a n d o f 1 3 c a r d s o u t o f a d e c k o f 5 2 c a r d s ?
/ p >
p c l a s s = ” c e n t e r ” > A n s w e r = s u b > 5 2 / s u b > C s u b > 1 3 / s u b > = 5 2 ! / ( 1 3 ! x 3 9 ! ) .
/ p >
p c l a s s = ” b o l d ” > E x e r c i s e 4 . 2 . 5 . / p >
p > F o u r f i n a n c i a l a w a r d s ( o f d i f f e r e n t v a l u e s ) w i l l b e g i v e n t o t h e ” b e s t ”
f o u r s t u d e n t s i n a c l a s s o f 1 6 3 . H o w m a n y p o s s i b l e w a y s c a n t h e s e a w a r d e e s
b e p i c k e d ? / p >
p > H e r e , t h e o r d e r c o u n t s . T h e a n s w e r i s s u b > 1 6 3 / s u b > P s u b > 4 / s u b > . / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 2 . 5 . / s p a n > H o w m a n y c o d e w o r d s o f l e n g t h
f o u r y o u c a n c o n s t r u c t o u t o f t h e E n g l i s h a l p h a b e t s ? T h i s p r o b l e m i s
n o t l i k e s e l e c t i n g 4 f r o m 2 6 l e t t e r s b e c a u s e w e c a n u s e t h e s a m e l e t t e r
m o r e t h a n o n c e . U s e t h e m u l t i p l i c a t i o n r u l e . / p >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > P i c k t h e 1 s t l e t t e r / t d >
t d > 2 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > P i c k t h e 2 n d l e t t e r / t d >
t d > 2 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 3 / t d >
t d > P i c k t h e 3 r d l e t t e r / t d >
t d > 2 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 4 / t d >
t d > P i c k t h e 4 t h l e t t e r / t d >
t d > 2 6 / t d >
/ t r >
/ t b o d y > / t a b l e >
p > T h e t o t a l n u m b e r o f s u c h w o r d s = 2 6 x 2 6 x 2 6 x 2 6 = 2 6 s u p > 4 / s u p > .
/ p >

h 3 > a n a m e = ” 4 ” > / a > 4 . 4 P r o b a b i l i t y S p a c e s w i t h E q u a l l y L i k e l y O u t c o m e s
a h r e f = ” # t o p ” > i m g s r c = ” . / i m a g e s / u p . g i f ” w i d t h = ” 2 0 ” h e i g h t = ” 1 3 ” a l t = ” G o t o t o p o f p a g e ” b o r d e r = ” 0 ” > / a >
/ h 3 >
p > U n l i k e t h e a b o v e p r o b l e m s , i n s o m e p r o b a b i l i t y s p a c e s e v e r y o u t c o m e
h a s a n e q u a l p r o b a b i l i t y . ( S u c h i s t h e c a s e w h e n y o u t o s s a ” n o r m a l ”
c o i n o r r o l l a ” n o r m a l ” d i e . ) I n s u c h c a s e s , w e s a y t h a t o u t c o m e s a r e
e q u a l l y l i k e l y . / p >
p > W h e n o u t c o m e s a r e e q u a l l y l i k e l y , t h e p r o b a b i l i t y P ( E ) c a n b e c o m p u t e d
b y c o u n t i n g t h e n u m b e r o f o u t c o m e s i n E a n d t h o s e o f S . I f S h a s N o u t c o m e s
t h e n w e h a v e : / p >
o l >
l i > P ( i n d i v i d u a l o u t c o m e ) = P ( o s u b > i / s u b > ) = 1 / N b r >
b r >
/ l i >
l i > F o r a n e v e n t E b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d >

s p a n c l a s s = ” b o l d ” > P ( E ) = ( # o f o u t c o m e s i n E ) / ( # o f o u t c o m e s
i n S ) &n b s p ; = &n b s p ; ( # o f o u t c o m e s i n E ) / N / s p a n >
/ t r >
/ t b o d y > / t a b l e >
b r >
/ l i >
l i > I f s p a n c l a s s = ” b o l d ” > n ( E ) / s p a n > = n u m b e r o f o u t c o m e s o n E , t h e n
b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d >
s p a n c l a s s = ” b o l d ” > P ( E ) = n ( E ) / n ( S ) &n b s p ; = &n b s p ; n ( E ) / N / s p a n >

/ t r >
/ t b o d y > / t a b l e >
/ l i >
/ o l >
p > s p a n c l a s s = ” b o l d ” > R e m a r k / s p a n > : O u t c o m e s a r e e q u a l l y l i k e l y i n ” n o r m a l ”
s i t u a t i o n s . E x a m p l e s a r e / p >
o l >
l i > t o s s i n g a n u n b i a s e d c o i n , / l i >
l i > t h r o w i n g a f a i r d i c e , / l i >
l i > p i c k i n g a c a r d f r o m a s h u f f l e d d e c k o f c a r d s a n d s o o n . / l i >
/ o l >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 4 . 1 . / s p a n > S u p p o s e w e r o l l a ( f a i r ) d i e
t h r e e t i m e s . / p >
o l >
l i > D e s c r i b e t h e s a m p l e s p a c e . / l i >
l i > C o u n t t h e n u m b e r o f o u t c o m e s i n t h e s a m p l e s p a c e S . / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t t h e s u m o f t h e p o i n t s o n t h e t h r e e
f a c e s i s 7 ? / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t t h e s u m o f t h e p o i n t s o n t h e f a c e s
i s n o t e q u a l t o 7 ? b r >
/ l i >
/ o l >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” w i d t h = ” 1 0 0 % ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > A n s w e r t o 1 i s b r >
s p a n c l a s s = ” b o l d ” > S = { ( 1 , 1 , 1 ) , ( 1 , 1 , 2 ) , ( 1 , 1 , 3 ) , ( 1 , 1 , 4 ) , ( 1 , 1 , 5 ) ,
( 1 , 2 , 6 ) , ( 1 , 2 , 1 ) , ( 1 , 2 , 2 ) , ( 1 , 2 , 3 ) &# 8 2 3 0 ; . } / s p a n > . b r >
b r >
A n o t h e r w a y t o w r i t e t h e s a m e i s
s p a n c l a s s = ” b o l d ” > S = { ( i , j , k ) : i , j , k = 1 , 2 , 3 , 4 , 5 , 6
} / s p a n > .
b r >
/ l i >
l i >
p > F i n d t h e t o t a l n u m b e r o f o u t c o m e s s p a n c l a s s = ” b o l d ” > N / s p a n >
i n S . b r >
T h e j o b o f r o l l i n g a d i e t h r e e t i m e s c a n b e d o n e i n t h r e e
s t a g e s : b r >
/ p >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > T h r o w t h e d i e / t d >
t d > 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > T h r o w t h e d i e 2 n d t i m e / t d >
t d > 6 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 3 / t d >
t d > T h r o w t h e d i e 3 r d t i m e / t d >
t d > 6 / t d >
/ t r >
/ t b o d y > / t a b l e >
p > S o , t h e t o t a l n u m b e r o f o u t c o m e s , b y m u l t i p l i c a t i o n p r i n c i p l e ,
b r >
i n S i s = s p a n c l a s s = ” b o l d ” > n ( S ) = N = 6 x 6 x 6 = 2 1 6 / s p a n > .
/ p >
/ l i >
l i > L e t E b e t h e e v e n t t h a t t h e s u m i s 7 . b r >
E = { ( 1 , 1 , 5 ) , ( 1 , 2 , 4 ) , ( 1 , 3 , 3 ) , ( 1 , 4 , 2 ) , ( 1 , 5 , 1 ) , b r >
( 2 , 1 , 4 ) , ( 2 , 2 , 3 ) , ( 2 , 3 , 2 ) , ( 2 , 4 , 1 ) , b r >
( 3 , 1 , 3 ) , ( 3 , 2 , 2 ) , ( 3 , 3 , 1 ) , b r >
( 4 , 1 , 2 ) , ( 4 , 2 , 1 ) , b r >
( 5 , 1 , 1 ) } b r >
S o E h a s n ( E ) = 1 5 o u t c o m e s . T h e r e f o r e , b r >
t h e p r o b a b i l i t y P ( E ) = n ( E ) / n ( S ) = 1 5 / 2 1 6 . b r >
b r >
/ l i >
l i > L e t F b e t h e e v e n t t h a t t h e s u m i s n o t e q u a l t o 7 . S o , t h e
n ( F ) = # o f o u t c o m e s i n F = ( # o f o u t c o m e s i n S ) – ( # o f o u t c o m e s
i n E ) = 2 1 6 – 1 5 = 2 0 1 . b r >
T h e r e f o r e b r >
P ( F ) = n ( F ) / n ( S ) = 2 0 1 / 2 1 6 . / l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 4 . 2 / s p a n > . S u p p o s e w e h a v e t o f o r m a
c o m m i t t e e o f t w o f r o m a g r o u p o f 1 5 m e n a n d 1 9 w o m e n . / p >
o l >
l i > D e s c r i b e t h e s a m p l e s p a c e a n d c o u n t t h e n u m b e r o f o u t c o m e s i n S .
/ l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t b o t h t h e m e m b e r s o f t h e c o m m i t t e e a r e
m e n ? / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t b o t h t h e m e m b e r s a r e w o m e n ? / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t o n e m e m b e r i s m a n a n d t h e o t h e r m e m b e r
i s w o m a n ? / l i >
/ o l >
H i n t : T h i s i s a p r o b l e m o f u n o r d e r e d – s e l e c t i o n . b r >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > S i s t h e s e t o f a l l p o s s i b l e p a i r s s e l e c t e d f r o m t h i s g r o u p
o f 1 5 + 1 9 = 3 4 p e o p l e . b r >
T h e n u m b e r o f o u t c o m e s i n S i s b r >
n ( S ) = N = s u b > 3 4 / s u b > C s u b > 2 / s u b > = s u b > 3 4 / s u b > P s u b > 2 / s u b > / 2 !
= ( 3 4 x 3 3 ) / 1 x 2 = 5 6 1 . / l i >
l i > L e t E b e t h e e v e n t t h a t b o t h t h e m e m b e r s o f t h e c o m m i t t e e
a r e m e n . T h e n u m b e r o f o u t c o m e s i n E = n ( E ) = n u m b e r o f w a y s
w e c a n s e l e c t t w o f r o m a g r o u p o f 1 5 m e n = s u b > 1 5 / s u b > C s u b > 2 / s u b > =
s u b > 1 5 / s u b > P s u b > 2 / s u b > / 2 ! = ( 1 5 x 1 4 ) / 2 = 1 0 5 . T h e r e f o r e
p > P ( E ) = n ( E ) / n ( S ) = 1 0 5 / 5 6 1 . / p >
/ l i >
l i > L e t F b e t h e e v e n t t h a t b o t h m e m b e r s o f t h e c o m m i t t e e a r e
w o m e n .
p > T h e n u m b e r o f o u t c o m e s i n F = n ( F ) = n u m b e r o f w a y s w e c a n
s e l e c t t w o f r o m a g r o u p o f 1 9 w o m e n = s u b > 1 9 / s u b > C s u b > 2 / s u b > =
s u b > 1 9 / s u b > P s u b > 2 / s u b > / 2 ! = ( 1 9 x 1 8 ) / 2 = 1 7 1 . T h e r e f o r e
/ p >
p > P ( F ) = n ( F ) / ( S ) = 1 7 1 / 5 6 1 . / p >
/ l i >
l i > L e t M b e t h e e v e n t t h a t o n e m e m b e r i s a m a n a n d t h e o t h e r
i s a w o m a n . T h e j o b o f p i c k i n g s u c h a c o m m i t t e e c a n b e d o n e
i n t w o s t a g e s : b r >
b r >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > P i c k a m a l e m e m b e r / t d >
t d > 1 5 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > P i c k a f e m a l e m e m b e r / t d >
t d > 1 9 / t d >
/ t r >
/ t b o d y > / t a b l e >
p > T h e n u m b e r o f o u t c o m e s i n M = n ( M ) = 1 5 x 1 9 = 2 8 5 . / p >
p > P ( M ) = n ( M ) / n ( S ) = 2 8 5 / 5 6 1 . / p >
/ l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
s p a n c l a s s = ” b o l d ” > P r o b l e m s o n 4 . 4 : C o u n t i n g a n d P r o b a b i l i t y / s p a n >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 1 / s p a n > . F i n d 5 ! b r >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 0 . h t m l ” >
S o l u t i o n / a > / p >
s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 2 / s p a n > .
p > S u p p o s e i n t h e W o r l d C u p S o c c e r t o u r n a m e n t , g r o u p A h a s e i g h t t e a m s .
N o w e a c h t e a m o f g r o u p A h a s t o p l a y w i t h a l l t h e o t h e r t e a m s i n t h e
g r o u p . F i n d h o w m a n y g a m e s w i l l b e p l a y e d a m o n g t h e G r o u p A t e a m s . / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 3 / s p a n > . H o w m a n y w a y s y o u c a n d e a l
a h a n d o f 1 3 c a r d s f r o m a d e c k o f 5 2 c a r d s ? / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 4 / s p a n > . H o w m a n y w a y s y o u c a n d e a l
a h a n d o f 4 S p a d e s , 3 H e a r t s , 3 D i a m o n d s , a n d 3 C l u b s ? b r >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 5 . h t m l ” > S o l u t i o n
/ a > b r >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 4 . h t m l ” > S o l u t i o n – v a r i a t i o n
/ a > / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 5 / s p a n > . W e h a v e 1 3 s t u d e n t s i n a
c l a s s . H o w m a n y w a y s w e c a n a s s i g n t h e f o u r s e a t s i n t h e f i r s t r o w ?
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 3 . h t m l ” > S o l u t i o n
/ a > / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 6 / s p a n > . P r o g r a m m i n g l a n g u a g e s s o m e t i m e s
u s e h e x a d e c i m a l s y s t e m ( a l s o c a l l e d ” h e x ” ) o f n u m b e r s . I n
t h i s s y s t e m 1 6 d i g i t s a r e u s e d a n d d e n o t e d b y 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , A , B , C , D , E , F .
S u p p o s e y o u f o r m a 6 – d i g i t n u m b e r i n h e x a d e c i m a l s y s t e m . / p >
o l >
l i > W h a t i s t h e p r o b a b i l i t y t h a t t h e n u m b e r w i l l s t a r t w i t h a l e t t e r – d i g i t ? / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t t h e n u m b e r i s d i v i s i b l e b y 1 6 ( i . e . ,
e n d s w i t h 0 ) ? / l i >
/ o l >

p > s p a n c l a s s = ” b o l d ” > S o l u t i o n / s p a n > b r >
H e r e t h e s a m p l e s p a c e i s t h e c o l l e c t i o n o f a l l t h e 6 – d i g i t h e x n u m b e r s .
/ p >
/ d i v >
u l >
l i > U s i n g t h e c o u n t i n g p r i n c i p l e , t h e n u m b e r o f h e x = n ( S ) = 1 6 s u p > 6 / s u p > .
/ l i >

l i > L e t E b e t h e e v e n t t h a t t h e n u m b e r s t a r t s w i t h a l e t t e r d i g i t . / l i >
l i > A g a i n , b y t h e c o u n t i n g p r i n c i p l e , t h e n u m b e r o f h e x i n E = n ( E ) =
6 * 1 6 s u p > 5 / s u p > . / l i >
l i > S o , P ( E ) = n ( E ) / n ( S ) = 6 / 1 6 . / l i >
l i > L e t F b e t h e e v e n t t h a t t h e n u m b e r i s d i v i s i b l e b y 1 6 . S i n c e a n u m b e r
i s d i v i s i b l e b y 1 6 m e a n s , i n h e x , t h e f i r s t d i g i t i s 0 . / l i >
l i > S o , t h e n u m b e r o f h e x i n F = n ( F ) = 1 6 s u p > 5 / s u p > * 1 = 1 6 s u p > 5 / s u p > . / l i >

l i > S o , P ( F ) = 1 6 s u p > 5 / s u p > / 1 6 s u p > 6 / s u p > = 1 / 1 6 . / l i >
/ u l >

p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 7 / s p a n > . Y o u a r e p l a y i n g b r i d g e , a n d
y o u a r e d e a l t a h a n d o f 1 3 c a r d s . / p >
o l >
l i > W h a t i s t h e p r o b a b i l i t y t h a t y o u w i l l g e t a h a n d o f 4 S p a d e s , 3
H e a r t s , 3 D i a m o n d s , a n d 3 C l u b s ? / l i >
l i > A l s o w h a t i s t h e p r o b a b i l i t y t h a t y o u w i l l g e t a l l t h e f o u r A c e s ?
/ l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t y o u w i l l g e t a l l 1 3 S p a d e s ? b r >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 6 . h t m l ” >
S o l u t i o n / a > a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 6 . h t m l ” > S o l u t i o n / a >
/ l i >
/ o l >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 8 / s p a n > . A c o m m i t t e e o f 9 i s s e l e c t e d
a t r a n d o m f r o m a g r o u p o f 1 1 s t u d e n t s , 1 7 m o t h e r s , a n d 1 3 f a t h e r s . / p >
o l >
l i > W h a t i s t h e p r o b a b i l i t y t h a t a l l t h e m e m b e r s o f t h e c o m m i t t e e a r e
s t u d e n t s ( i . e . , a c o m m i t t e e w i t h o u t a n y e x p e r i e n c e ) ? / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t t h e c o m m i t t e e h a s 3 s t u d e n t s , 3 m o t h e r s ,
a n d 3 f a t h e r s ( i . e . , a b a l a n c e d c o m m i t t e e ) ? b r >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 7 . h t m l ” > S o l u t i o n
/ a > b r >
/ l i >
/ o l >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 4 . 9 / s p a n > . T h r e e s c h o l a r s h i p s o f u n e q u a l
v a l u e s w i l l h a v e t o b e a w a r d e d t o a g r o u p o f 3 5 a p p l i c a n t s . H o w m a n y
w a s s u c h a s e l e c t i o n c a n b e m a d e ? a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 1 2 . h t m l ” > S o l u t i o n / a >
/ p >
h 3 > a n a m e = ” 5 ” > / a > 4 . 5 T h e C o m p l e m e n t o f a n E v e n t a h r e f = ” # t o p ” > i m g s r c = ” . / i m a g e s / u p . g i f ” w i d t h = ” 2 0 ” h e i g h t = ” 1 3 ” a l t = ” G o t o t o p o f p a g e ” b o r d e r = ” 0 ” > / a > / h 3 >
p > S o m e t i m e s i t i s e a s i e r t o f i n d t h e p r o b a b i l i t y t h a t a n e v e n t E d o e s
n o t o c c u r . T h e n w e c a n u s e t h i s t o c o m p u t e t h e p r o b a b i l i t y t h a t E o c c u r s . / p >
o l >
l i > s p a n c l a s s = ” b o l d ” > D e f i n i t i o n / s p a n > : S u p p o s e S i s t h e s a m p l e s p a c e ,
a n d E i s a n e v e n t . T h e n ( i > s p a n c l a s s = ” r e d ” > n o t / s p a n > / i > s p a n c l a s s = ” r e d ” >
E / s p a n > ) w i l l d e n o t e t h e e v e n t t h a t E d o e s n o t o c c u r . ( n o t E ) i s
a l s o c a l l e d s p a n c l a s s = ” r e d ” > t h e c o m p l e m e n t o f E / s p a n > o r s p a n c l a s s = ” r e d ” > t h e
o p p o s i t e / s p a n > e v e n t o f E . / l i >
l i > s p a n c l a s s = ” b o l d ” > F o r m u l a / s p a n > : W e h a v e b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d a l i g n = ” c e n t e r ” > s p a n c l a s s = ” b o l d r e d ” > P ( E ) + P ( n o t E ) = 1 / s p a n > b r >
O r b r >
s p a n c l a s s = ” b o l d r e d ” > P ( E ) = 1 – P ( n o t E ) / s p a n > . / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 5 . 1 / s p a n > . S u p p o s e w e r o l l a d i e
t h r e e t i m e s . W h a t i s t h e p r o b a b i l i t y t h a t t h e f a c e 6 w i l l s h o w u p
a t l e a s t o n c e ? / p >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” b g c o l o r = ” # f f f f f f ” >
t b o d y > t r a l i g n = ” c e n t e r ” s t y l e = ” b a c k g r o u n d – c o l o r : r g b ( 1 5 3 , 1 5 3 , 1 5 3 ) ; ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > L e t E b e t h e e v e n t t h a t t h e f a c e 6 w i l l s h o w u p a t l e a s t
o n c e . / l i >
l i > W e h a v e s e e n t h a t t h e s a m p l e s p a c e S h a s 6 s u p > 3 / s u p >
o u t c o m e s . / l i >
l i > W e n e e d t o c o m p u t e t h e n u m b e r n ( E ) o f o u t c o m e s i n E .
B u t i t w i l l b e e a s i e r t o c o u n t t h e n u m b e r o f o u t c o m e s i n
( n o t E ) . / l i >
l i > ( n o t E ) i s t h e e v e n t t h a t t h e f a c e 6 n e v e r s h o w e d u p
i n t h e t h r e e r o l l s . T h i s c a n b e a c h i e v e d i n t h e f o l l o w i n g
t h r e e s t a g e s : b r >
b r >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r a l i g n = ” c e n t e r ” b g c o l o r = ” # 9 9 9 9 9 9 ” >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > R o l l 1 – 5 i n 1 s t t h r o w / t d >
t d > 5 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > R o l l 1 – 5 i n 2 n d t h r o w / t d >
t d > 5 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 3 / t d >
t d > R o l l 1 – 5 i n 3 r d t h r o w / t d >
t d > 5 / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
/ l i >
l i > T h e n u m b e r o f o u t c o m e s i n ( n o t E ) = n ( n o t E ) = 5 s u p > 3 / s u p > =
1 2 5 . / l i >
l i > T h e p r o b a b i l i t y P ( n o t E ) = n ( n o t E ) / n ( S ) = 1 2 5 / 2 1 6 . / l i >
l i > s p a n c l a s s = ” b o l d ” > P ( E ) = 1 – P ( n o t E ) = 1 – 1 2 5 / 2 1 6
= 9 1 / 2 1 6 . / s p a n > / l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 5 . 2 / s p a n > . S u p p o s e w e r o l l a d i e
s e v e n t i m e s . W h a t i s t h e p r o b a b i l i t y t h a t a n e v e n – n u m b e r f a c e w i l l
s h o w u p a t l e a s t o n c e ? / p >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” s t y l e = ” b a c k g r o u n d – c o l o r : r g b ( 1 5 3 , 1 5 3 , 1 5 3 ) ; ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > T h e e x p e r i m e n t o f r o l l i n g a d i e s e v e n t i m e s c a n b e p e r f o r m e d
i n ” 7 s t a g e s . ” / l i >
l i > B y t h e m u l t i p l i c a t i o n r u l e t h e s a m p l e s p a c e S h a s n ( S )
= N = 6 s u p > 7 / s u p > o u t c o m e s . / l i >
l i > J u s t r e m e m b e r t h a t t h e s p a n c l a s s = ” r e d ” > o p p o s i t e o f ” a t
l e a s t o n c e ” i s ” n e v e r . ” / s p a n > / l i >
l i > L e t u s d e n o t e b y E t h e e v e n t t h a t a n e v e n – n u m b e r f a c e
w i l l s p a n c l a s s = ” r e d ” > s h o w u p a t l e a s t o n c e / s p a n > . / l i >
l i > ( n o t E ) i s t h e e v e n t t h a t a n e v e n – n u m b e r f a c e n e v e r s h o w s
u p . / l i >
l i > A n o u t c o m e i n ( n o t E ) c a n b e a c c o m p l i s h e d i n t h e f o l l o w i n g
s e v e n s t a g e s : b r >
b r >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” >
t b o d y > t r a l i g n = ” c e n t e r ” >
t h w i d t h = ” 3 8 ” > S t a g e / t h >
t h w i d t h = ” 1 3 2 ” > T h e j o b / t h >
t h w i d t h = ” 1 1 7 ” > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 1 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 1 s t r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 2 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 2 n d r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 3 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 3 r d r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 4 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 4 t h r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 5 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 5 t h r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 6 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 6 t h r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d w i d t h = ” 3 8 ” > 7 / t d >
t d w i d t h = ” 1 3 2 ” > R o l l 1 , 3 , 5 i n 7 t h r o l l / t d >
t d w i d t h = ” 1 1 7 ” > 3 / t d >
/ t r >
/ t b o d y > / t a b l e >
/ l i >
l i > T h e n u m b e r o f o u t c o m e s i n ( n o t E ) = n ( n o t E ) = 3 s u p > 7 / s u p > .
/ l i >
l i > T h e p r o b a b i l i t y s p a n c l a s s = ” b o l d ” > P ( n o t E ) = n ( n o t E ) / n ( S )
= 3 s u p > 7 / s u p > / 6 s u p > 7 / s u p > / s p a n > . / l i >
l i > s p a n c l a s s = ” b o l d ” > P ( E ) = 1 – P ( n o t E ) = 1 – 3 s u p > 7 / s u p > / 6 s u p > 7 / s u p > / s p a n > .
/ l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e > / l i > / o l >
p c l a s s = ” b o l d ” > P r o b l e m s o n 4 . 5 : T h e C o m p l e m e n t o f a n E v e n t / p >

s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 5 . 1 / s p a n > . A p e r s o n o w n s t w o s t o c k s . T h e
p r o b a b i l i t y o f t h e e v e n t t h a t a t l e a s t o n e s t o c k w i l l g o u p i n p r i c e o n
a p a r t i c u l a r d a y i s P ( E ) = 0 . 9 . W h a t i s t h e p r o b a b i l i t y t h a t n o n e w i l l
g o u p i n p r i c e o n a p a r t i c u l a r d a y ? s p a n c l a s s = ” b o l d ” > A n s w e r : P ( n o t E )
= 1 – 0 . 9 = . 0 1 . / s p a n >
h 3 > a n a m e = ” 6 ” > / a > 4 . 6 I n d e p e n d e n t E v e n t s a h r e f = ” # t o p ” > i m g s r c = ” . / i m a g e s / u p . g i f ” w i d t h = ” 2 0 ” h e i g h t = ” 1 3 ” a l t = ” G o t o t o p o f p a g e ” b o r d e r = ” 0 ” > / a > / h 3 >
p > S o m e t i m e s i t i s u n d e r s t a n d a b l e t h a t t w o e v e n t s E a n d F d o n o t i n f l u e n c e
t h e o c c u r r e n c e o f e a c h o t h e r . F o r e x a m p l e , i f y o u r o l l a d i e t w i c e a n d
E i s t h e e v e n t t h a t t h e s p a n c l a s s = ” r e d ” > f i r s t r o l l / s p a n > w i l l s h o w
a n o d d – n u m b e r f a c e a n d F i s t h e e v e n t t h a t t h e s p a n c l a s s = ” r e d ” > s e c o n d
r o l l / s p a n > w i l l s h o w 1 o r 2 , t h e n i t i s r e a s o n a b l e t o a s s u m e t h a t t h e
o c c u r r e n c e o f E w i l l n o t i n f l u e n c e t h e o c c u r r e n c e o f F . ( D e s c r i b e E
a n d F i n b r a c e n o t a t i o n . ) / p >
o l >
l i > s p a n c l a s s = ” b o l d ” > D e f i n i t i o n / s p a n > : W e s a y t h a t t h e t w o e v e n t s
E a n d F a r e m u t u a l l y s p a n c l a s s = ” r e d ” > i n d e p e n d e n t / s p a n > i f t h e o c c u r r e n c e
o f o n e d o e s n o t i n f l u e n c e t h e o c c u r r e n c e o f t h e o t h e r . / l i >
l i > s p a n c l a s s = ” b o l d ” > T h e M u l t i p l i c a t i o n P r i n c i p l e o f I n d e p e n d e n c e / s p a n > :
S u p p o s e E a n d F a r e t w o i n d e p e n d e n t e v e n t s . T h e n t h e p r o b a b i l i t y t h a t
b o t h E a n d F o c c u r i s t h e p r o d u c t P ( E ) P ( F ) . / l i >
l i > I f E a n d F a r e i n d e p e n d e n t , t h e n b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d > s p a n c l a s s = ” b o l d r e d ” > P ( E a n d F ) = P ( E ) P ( F ) / s p a n > . / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
/ l i >
/ o l >
N o w w e a r e g o i n g t o u s e t h i s m u l t i p l i c a t i o n p r i n c i p l e t o c o m p u t e s o m e
p r o b a b i l i t i e s .
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 6 . 1 . / s p a n > S u p p o s e y o u a r e d e a l t a h a n d
o f f i v e c a r d s o u t o f a s h u f f l e d d e c k o f t w e n t y h i g h – c a r d s . ( A c e , K i n g ,
Q u e e n , J a c k , a n d 1 0 a r e t h e h i g h – c a r d s . ) / p >
o l >
l i > W h a t i s t h e p r o b a b i l i t y t h a t y o u w i l l r e c e i v e a l l f o u r a c e s ? / l i >
l i > S u p p o s e y o u a r e d e a l t s u c h a h a n d o f f i v e c a r d s t w i c e . W h a t i s t h e
p r o b a b i l i t y t h a t y o u w i l l r e c e i v e a l l t h e f o u r a c e s i n b o t h t h e d e a l s ?
/ l i >
/ o l >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” b g c o l o r = ” # 9 9 9 9 9 9 ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > H e r e t h e r a n d o m e x p e r i m e n t i s t o d e a l a h a n d o f f i v e c a r d s
o u t o f t w e n t y h i g h – c a r d s . T h e s a m p l e s p a c e i s t h e s e t o f a l l
p o s s i b l e c o m b i n a t i o n s o f t w e n t y c a r d s t a k e n f i v e a t a t i m e .
b r >
/ l i >
l i > T h e t o t a l n u m b e r o f o u t c o m e s i n S i s = b r >
s p a n c l a s s = ” b o l d ” > N = n ( S ) = s u b > 2 0 / s u b > C s u b > 5 / s u b > = s u b > 2 0 / s u b > P s u b > 5 / s u b > / 5 ! =
( 2 0 x 1 9 x 1 8 x 1 7 x 1 6 ) / ( 1 x 2 x 3 x 4 x 5 ) = 1 5 5 0 4 / s p a n > .
b r >
/ l i >
l i > N o w l e t E b e t h e e v e n t t h a t a h a n d o f f i v e c a r d s h a s a l l f o u r
a c e s . S u c h a h a n d c o u l d b e d e a l t i n t w o s t a g e s : b r >
b r >
t a b l e b o r d e r = ” 1 ” a l i g n = ” c e n t e r ” w i d t h = ” 4 2 5 ” >
t b o d y > t r >
t h > S t a g e / t h >
t h > T h e j o b / t h >
t h > N u m b e r o f w a y s / t h >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 1 / t d >
t d > P i c k t h e 4 a c e s f r o m 4 / t d >
t d > s u b > 4 / s u b > C s u b > 4 / s u b > = 1 / t d >
/ t r >
t r a l i g n = ” c e n t e r ” >
t d > 2 / t d >
t d > P i c k 1 m o r e c a r d f r o m t h e r e m a i n i n g 1 6 / t d >
t d > s u b > 1 6 / s u b > C s u b > 1 / s u b > = 1 6 / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
/ l i >
l i > B y m u l t i p l i c a t i o n p r i n c i p l e , t h e n u m b e r o f o u t c o m e s i n s p a n c l a s s = ” b o l d ” > E
= n ( E ) = 1 x 1 6 = 1 6 / s p a n > . b r >
/ l i >
l i > s p a n c l a s s = ” b o l d ” > P ( E ) = n ( E ) / n ( S ) = 1 6 / 1 5 5 0 4 / s p a n > . / l i >
/ o l >
b r >
N o w w e p r o c e e d t o s o l v e t h e s e c o n d p a r t : b r >
o l >
l i > N o w o u r e x p e r i m e n t i s t o d e a l a h a n d o f f i v e c a r d s o u t o f
t w e n t y c a r d s t w i c e . b r >
/ l i >
l i > L e t W b e t h e e v e n t t h a t w e g e t a l l f o u r a c e s t w i c e . b r >
/ l i >
l i > L e t E s u b > 1 / s u b > b e t h e e v e n t t h a t w e g e t a l l f o u r a c e s i n
t h e f i r s t d e a l a n d E s u b > 2 / s u b > b e t h e e v e n t t h a t w e g e t a l l
f o u r a c e s i n t h e s e c o n d d e a l . b r >
/ l i >
l i > W = ( E s u b > 1 / s u b > a n d E s u b > 2 / s u b > ) . b r >
/ l i >
l i > F r o m t h e f i r s t p a r t , w e h a v e
o l t y p e = ” a ” >
l i > P ( E s u b > 1 / s u b > ) = 1 6 / 1 5 5 0 4 . / l i >
l i > P ( E s u b > 2 / s u b > ) = 1 6 / 1 5 5 0 4 . / l i >
/ o l >
/ l i >
l i > A l s o E s u b > 1 / s u b > a n d E s u b > 2 / s u b > a r e m u t u a l l y i n d e p e n d e n t .
b r >
/ l i >
l i > B y t h e m u l t i p l i c a t i o n r u l e o f i n d e p e n d e n t e v e n t s , w e h a v e
s p a n c l a s s = ” b o l d ” > P ( W ) = P ( E s u b > 1 / s u b > ) P ( E s u b > 2 / s u b > ) =
( 1 6 / 1 5 5 0 4 ) s u p > 2 / s u p > / s p a n > . / l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 6 . 2 / s p a n > . S u p p o s e y o u r o l l f o u r f a i r
d i c e . / p >
o l t y p e = ” a ” >
l i > W h a t i s t h e p r o b a b i l i t y t h a t t h e f a c e 3 w i l l n e v e r a p p e a r ? / l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t a t l e a s t o n e d i e w i l l s h o w t h e f a c e
3 ? / l i >
/ o l >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” w i d t h = ” 1 0 0 % ” >
t b o d y > t r a l i g n = ” c e n t e r ” s t y l e = ” b a c k g r o u n d – c o l o r : r g b ( 1 5 3 , 1 5 3 , 1 5 3 ) ; ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > H e r e t h e s a m p l e s p a c e S h a s N = n ( S ) = 6 s u p > 4 / s u p > o u t c o m e s .
b r >
/ l i >
l i > L e t E b e t h e e v e n t t h a t t h e f a c e 3 w i l l n e v e r a p p e a r . b r >
/ l i >
l i > E i s t h e e v e n t t h a t n o n e o f t h e f o u r d i c e w i l l s h o w t h e f a c e
3 . b r >
/ l i >
l i > S u p p o s e F s u b > 1 / s u b > i s t h e e v e n t t h a t t h e f i r s t
d i e w i l l n o t s h o w t h e f a c e 3 . / l i >
l i > S u p p o s e F s u b > 2 / s u b > i s t h e e v e n t t h a t t h e s e c o n d
d i e w i l l n o t s h o w t h e f a c e 3 . / l i >
l i > S u p p o s e F s u b > 3 / s u b > i s t h e e v e n t t h a t t h e t h i r d
d i e w i l l n o t s h o w t h e f a c e 3 . / l i >
l i > S u p p o s e F s u b > 4 / s u b > i s t h e e v e n t t h a t t h e f o u r t h
d i e w i l l n o t s h o w t h e f a c e 3 . b r >
/ l i >
l i > I t i s r e a s o n a b l e t o a s s u m e t h a t F s u b > 1 / s u b > , F s u b > 2 / s u b > ,
F s u b > 3 / s u b > , a n d F s u b > 4 / s u b > a r e m u t u a l l y i n d e p e n d e n t . b r >
/ l i >
l i > A l s o , E = ( F s u b > 1 / s u b > a n d F s u b > 2 / s u b > a n d F s u b > 3 / s u b >
a n d F s u b > 4 / s u b > ) . b r >
/ l i >
l i > W e a l s o h a v e b r >
p a l i g n = ” c e n t e r ” > s p a n c l a s s = ” b o l d ” > P ( F s u b > 1 / s u b > ) = P ( F s u b > 2 / s u b > )
= P ( F s u b > 3 / s u b > ) = P ( F s u b > 4 / s u b > ) = 5 / 6 . / s p a n > / p >
/ l i >
l i > s p a n c l a s s = ” b o l d ” > P ( E ) / s p a n > b > = / b > s p a n c l a s s = ” b o l d ” > P ( F s u b > 1 / s u b > )
x P ( F s u b > 2 / s u b > ) x P ( F s u b > 3 / s u b > ) x P ( F s u b > 4 / s u b > ) =
( 5 / 6 ) s u p > 4 / s u p > / s p a n > . b r >
T h e p r o b a b i l i t y t h a t t h e f a c e 3 w i l l n e v e r a p p e a r i s ( 5 / 6 ) s u p > 4 / s u p > . / l i >
/ o l >
N o w w e s o l v e ( b ) : b r >
o l >
l i > L e t F b e t h e e v e n t t h a t f a c e 3 w i l l s h o w u p a t l e a s t o n c e .
b r >
/ l i >
l i > B e c a u s e t h e o p p o s i t e o f ” a t l e a s t o n c e ” i s ” n e v e r , ” w e h a v e
s p a n c l a s s = ” b o l d ” > F = ( n o t E ) / s p a n > . b r >
/ l i >
l i > S o , P ( F ) = 1 – P ( E ) = 1 – ( 5 / 6 ) s u p > 4 / s u p > . / l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
p c l a s s = ” b o l d ” > P r o b l e m s o n 4 . 6 : I n d e p e n d e n t E v e n t s / p >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 6 . 1 / s p a n > . T w o u n i v e r s i t y e m p l o y e e s
( M r . P a r k a n d M r . J o n e s ) i s s u e t i c k e t s t o i l l e g a l l y p a r k e d c a r s . P r o b a b i l i t y
o f t h e e v e n t E t h a t M r . J o n e s w i l l n o t i c e a n i l l e g a l l y p a r k e d c a r i s
P ( E ) = 0 . 1 , a n d t h e p r o b a b i l i t y o f t h e e v e n t F t h a t M r . P a r k w i l l n o t i c e
a n i l l e g a l l y p a r k e d c a r i s P ( F ) = 0 . 3 . / p >
o l >
l i > W h a t i s t h e p r o b a b i l i t y P ( n o t E ) t h a t M r . J o n e s w i l l m i s s a n i l l e g a l l y
p a r k e d c a r ? b r >
A n s w e r : P ( n o t E ) = 1 – P ( E ) = 1 – 0 . 1 = 0 . 9 . / l i >
l i > W h a t i s t h e p r o b a b i l i t y P ( n o t F ) t h a t M r . P a r k w i l l m i s s a n i l l e g a l l y
p a r k e d c a r ? b r >
A n s w e r : P ( n o t F ) = 1 – P ( F ) = 1 – 0 . 3 = 0 . 7 . / l i >
l i > A s s u m i n g i n d e p e n d e n c e , w h a t i s t h e p r o b a b i l i t y t h a t b o t h w i l l m i s s
a n i l l e g a l l y p a r k e d c a r ? b r >
A n s w e r : P ( ( n o t E ) a n d ( n o t F ) ) = P ( n o t E ) * P ( n o t F ) = 0 . 9 * 0 . 7 = 0 . 6 3 .
/ l i >
l i > W h a t i s t h e p r o b a b i l i t y t h a t a t l e a s t o n e o f t h e m w i l l n o t i c e a n
i l l e g a l l y p a r k e d c a r ? b r >
A n s w e r : P ( a t l e a s t o n e w i l l n o t i c e ) = 1 – P ( B o t h w i l l m i s s ) = 1 – . 6 3 .
/ l i >
/ o l >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 6 . 2 / s p a n > . S u p p o s e t h e t w o e n g i n e s o f
a a i r p l a n e f u n c t i o n i n d e p e n d e n t l y . P r o b a b i l i t y t h a t t h e f i r s t e n g i n e
f a i l s i n a f l i g h t i s . 0 1 , a n d t h e p r o b a b i l i t y t h a t t h e s e c o n d e n g i n e
f a i l s i n a f l i g h t i s . 0 2 . / p >
o l >
l i > W h a t i s t h e p r o b a b i l i t y t h a t b o t h w i l l f a i l i n a f l i g h t ? / l i >
l i > W h a t a r e t h e o d d s i n f a v o r o f b o t h e n g i n e s f a i l i n g i n a f l i g h t ?
/ l i >
/ o l >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 1 0 5 / m 1 0 5 f l a s h / t o p i c C 4 p 1 . h t m l ” > S o l u t i o n / a >
p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 6 . 3 / s p a n > . T h e p r o b a b i l i t y t h a t y o u
w i l l r e c e i v e a w r o n g n u m b e r c a l l t h i s w e e k i s 0 . 3 , a n d t h e p r o b a b i l i t y
t h a t y o u w i l l r e c e i v e a s a l e s c a l l t h i s w e e k i s 0 . 8 , a n d t h e p r o b a b i l i t y
t h a t y o u w i l l r e c e i v e a s u r v e y c a l l t h i s w e e k i s 0 . 5 . W h a t i s t h e p r o b a b i l i t y
t h a t y o u w i l l r e c e i v e o n e o f e a c h t h i s w e e k ? ( A s s u m e i n d e p e n d e n c e . )
b r >
a h r e f = ” h t t p : / / w w w . m a t h . k u . e d u / % 7 E m a n d a l / m a t h 3 6 5 / m 3 6 5 f l a s h / s t a t C 3 p 2 2 . h t m l ” >
S o l u t i o n / a > b r >
/ p >

p > s p a n c l a s s = ” b o l d ” > E x e r c i s e 4 . 6 . 4 / s p a n > .
T h e p r o b a b i l i t y , o f t h e e v e n t E ,
t h a t a s t u d e n t w i l l m a j o r i n e i t h e r l i b e r a l a r t s
o r i n b u s i n e s s i s P ( E ) = . 6 9 . F i n d t h e p r o b a b i l i t y
t h a t t h e s t u d e n t w i l l m a j o r n e i t h e r i n l i b e r a l a r t s
n o r i n b u s i n e s s .
b r >
A n s w e r : P ( n o t E ) = 1 – P ( E ) = 1 – . 6 9 = 3 1 . / p >

h 3 > a n a m e = ” 7 ” > / a > 4 . 7 O d d s f o r a n d a g a i n s t a h r e f = ” # t o p ” > i m g s r c = ” . / i m a g e s / u p . g i f ” w i d t h = ” 2 0 ” h e i g h t = ” 1 3 ” a l t = ” G o t o t o p o f p a g e ” b o r d e r = ” 0 ” > / a > / h 3 >
p > I n m a n y s i t u a t i o n s t h e p r o b a b i l i t y o f a n e v e n t E i s d e s c r i b e d a s ” o d d s ”
i n f a v o r o r a g a i n s t . T h i s l a n g u a g e i s o f t e n u s e d i n g a m b l i n g , h o r s e
r a c e s , a n d s p o r t s . / p >
o l >
l i > s p a n c l a s s = ” b o l d ” > D e f i n i t i o n / s p a n > . S u p p o s e E i s a n e v e n t . W e
s a y t h a t t h e s p a n c l a s s = ” r e d ” > o d d s i n f a v o r / s p a n > o f E i s ” m t o
n ” t o m e a n t h a t b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d > s p a n c l a s s = ” b o l d r e d ” > P ( E ) = m / ( m + n ) / s p a n > . / t d >
/ t r >
/ t b o d y > / t a b l e >
b r >
I f P ( E ) = a / b t h e n t h e s p a n c l a s s = ” b o l d ” > o d d s i n f a v o r o f E i s ” a
t o b – a . ” / s p a n > b r >
b r >
/ l i >
l i > s p a n c l a s s = ” b o l d ” > D e f i n i t i o n / s p a n > . W e s a y t h a t s p a n c l a s s = ” r e d ” > t h e
o d d s a g a i n s t / s p a n > t h e e v e n t E i s ” n t o m , ” i f t h e o d d s i n f a v o r
o f E i s ” m t o n . ” O d d s a g a i n s t E i s ” n t o m ” i f b r >
t a b l e a l i g n = ” c e n t e r ” >
t b o d y > t r >
t d > s p a n c l a s s = ” b o l d r e d ” > P ( n o t E ) = m / ( m + n ) / s p a n > . / t d >
/ t r >
/ t b o d y > / t a b l e >
/ l i >
/ o l >
b r >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 7 . 1 / s p a n > . S u p p o s e y o u r o l l a d i e t w i c e .
/ p >
o l t y p e = ” a ” >
l i > W h a t i s t h e p r o b a b i l i t y o f t h e e v e n t E t h a t a t l e a s t o n e o f t h e
t w o r o l l s w i l l s h o w 4 ? b r >
/ l i >
l i > W h a t a r e t h e o d d s i n f a v o r o f t h e e v e n t E ? / l i >
/ o l >
b r >
b r >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” s t y l e = ” b a c k g r o u n d – c o l o r : r g b ( 1 5 3 , 1 5 3 , 1 5 3 ) ; ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l >
l i > F i r s t w e c o m p u t e P ( n o t E ) . / l i >
l i > B u t ( n o t E ) i s t h e e v e n t t h a t t h e f a c e 4 d i d n o t s h o w u p
i n t h e s e t w o r o l l s . / l i >
l i > L e t F s u b > 1 / s u b > b e t h e e v e n t t h a t f a c e 4 d i d n o t s h o w u p
i n t h e f i r s t r o l l . / l i >
l i > L e t F s u b > 2 / s u b > b e t h e e v e n t t h a t f a c e 4 d i d n o t s h o w u p
i n t h e s e c o n d r o l l . / l i >
l i > T o c o m p u t e P ( F s u b > 1 / s u b > ) w e j u s t h a v e t o l o o k a t t h e f i r s t
r o l l . P ( F s u b > 1 / s u b > ) = b r >
( # o f o u t c o m e s i n f i r s t r o l l t h a t i s n o t 4 ) / ( # o f o u t c o m e s i n
t h e f i r s t r o l l ) = 5 / 6 . / l i >
l i > S i m i l a r l y , P ( F s u b > 2 / s u b > ) = 5 / 6 . / l i >
l i > A l s o ( n o t E ) = ( F s u b > 1 / s u b > a n d F s u b > 2 / s u b > ) . / l i >
l i > P ( n o t E ) = P ( F s u b > 1 / s u b > a n d F s u b > 2 / s u b > ) = P ( F s u b > 1 / s u b > )
P ( F s u b > 2 / s u b > ) = ( 5 / 6 ) ( 5 / 6 ) = 2 5 / 3 6 . / l i >
l i > P ( E ) = 1 – P ( n o t E ) = 1 – 2 5 / 3 6 = 1 1 / 3 6 . / l i >
l i > P ( E ) = 1 1 / 3 6 . / l i >
l i > W e h a v e o d d s i n f a v o r o f E i s 1 1 t o ( 3 6 – 1 1 ) . T h e o d d s i n
f a v o r o f t h e e v e n t E i s 1 1 t o 2 5 . / l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
p > s p a n c l a s s = ” b o l d ” > E x a m p l e 4 . 7 . 2 / s p a n > . S u p p o s e w e h a v e a g a m e w h e r e
w e r o l l t w o d i c e . W e w i n i f t h e s u m o f t h e ” f a c e v a l u e s ” i s l e s s o r
e q u a l t o f i v e ; o t h e r w i s e w e l o s e . / p >
o l >
l i > F i n d t h e p r o b a b i l i t y o f w i n n i n g . A l s o w h a t i s t h e o d d s i n f a v o r
o f w i n n i n g ? / l i >
l i > I f w e r o l l t h e d i c e t w i c e ( i . e . , w e p l a y t w i c e ) , w h a t i s t h e p r o b a b i l i t y
t h a t w e w i n i n b o t h t h e r o l l s ? / l i >
l i > I f w e r o l l t h e d i c e t w i c e ( i . e . , w e p l a y t w i c e ) , w h a t i s t h e p r o b a b i l i t y
t h a t w e l o s e i n b o t h t h e r o l l s ? b r >
b r >
/ l i >
/ o l >
t a b l e a l i g n = ” c e n t e r ” b o r d e r = ” 1 ” >
t b o d y > t r a l i g n = ” c e n t e r ” s t y l e = ” b a c k g r o u n d – c o l o r : r g b ( 1 5 3 , 1 5 3 , 1 5 3 ) ; ” >
t h > T h e S o l u t i o n / t h >
/ t r >
t r >
t d >
o l t y p e = ” a ” >
l i > H e r e t h e e x p e r i m e n t i s t o r o l l t w o d i c e . T h e s a m p l e s p a c e
h a s N = n ( S ) = 3 6 o u t c o m e s . / l i >
l i > L e t E b e t h e e v e n t t h a t y o u w i n . / l i >
l i > T h e n E = { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) ,
( 3 , 1 ) , ( 3 , 2 ) , ( 4 , 1 ) } . / l i >
l i > E h a s n ( E ) = 1 0 o u t c o m e s i n i t . / l i >
l i > s p a n c l a s s = ” b o l d ” > P ( E ) = n ( E ) / n ( S ) = 1 0 / 3 6 / s p a n > . / l i >
l i > T h e o d d s i n f a v o r o f w i n n i n g a r e 1 0 t o 2 6 ( = 3 6 – 1 0 ) .
p > / p >
/ l i >
/ o l >
P r o b a b i l i t y o f w i n n i n g t w i c e :
o l >
l i > L e t E s u b > 1 / s u b > b e t h e e v e n t t h a t y o u w i n i n t h e f i r s t
t i m e a n d E s u b > 2 / s u b > b e t h e e v e n t t h a t y o u w i n i n t h e s e c o n d
t i m e . / l i >
l i > L e t W b e t h e e v e n t t h a t y o u w i n b o t h t h e r o l l s . / l i >
l i > T h e n s p a n c l a s s = ” b o l d ” > W = ( E s u b > 1 / s u b > a n d E s u b > 2 / s u b > ) / s p a n > .
/ l i >
l i > A n d s p a n c l a s s = ” b o l d ” > P ( W ) = P ( E s u b > 1 / s u b > a n d E s u b > 2 / s u b > )
= P ( E s u b > 1 / s u b > ) P ( E s u b > 2 / s u b > ) = ( 1 0 / 3 6 ) ( 1 0 / 3 6 ) = 1 0 0 / 1 2 9 6 / s p a n > .
/ l i >
l i > P r o b a b l i l i t y o f l o s i n g t w i c e :
o l >
l i > L e t L s u b > 1 / s u b > b e t h e e v e n t t h a t y o u l o s e t h e f i r s t
t i m e a n d L s u b > 2 / s u b > b e t h e e v e n t t h a t y o u l o s e t h e s e c o n d
t i m e . / l i >
l i > L e t L b e t h e e v e n t t h a t y o u l o s e b o t h t h e r o l l s . / l i >
l i > T h e n L = ( L s u b > 1 / s u b > a n d L s u b > 2 / s u b > ) . / l i >
l i > A n d P ( L ) = P ( L s u b > 1 / s u b > a n d L s u b > 2 / s u b > ) = P ( L s u b > 1 / s u b > )
P ( L s u b > 2 / s u b > ) . / l i >
l i > W e a l s o h a v e P ( L s u b > 1 / s u b > ) = 1 – P ( E s u b > 1 / s u b > ) =
1 – 1 0 / 3 6 = 2 6 / 3 6 . / l i >
l i > A n d s i m i l a r l y P ( L s u b > 2 / s u b > ) = 2 6 / 3 6 . / l i >
l i > P ( L ) = P ( L s u b > 1 / s u b > ) P ( L s u b > 2 / s u b > ) = ( 2 6 / 3 6 ) ( 2 6 / 3 6 ) .
/ l i >
/ o l >
/ l i >
/ o l >
/ t d >
/ t r >
/ t b o d y > / t a b l e >
p c l a s s = ” b o l d ” > b r >
/ p >
p > a h r e f = ” # t o p ” > b a c k t o t o p / a > a h r e f = ” # t o p ” > i m g s r c = ” . / i m a g e s / u p . g i f ” w i d t h = ” 2 0 ” h e i g h t = ” 1 3 ” a l t = ” G o t o t o p o f p a g e ” b o r d e r = ” 0 ” > / a > / p >
/ d i v >

/ t d >
/ t r >
/ t b o d y > / t a b l e >

/ b o d y > / h t m l >

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