1
Chapter 5
Probability
 Probability
o The probability of an outcome is defined as the long-term proportion
of times the outcome occurs.
 Building Blocks of Probability
o An experiment is any activity for which the outcome in uncertain.
o An outcome is the result of a single performance of an experiment.
o The collection of all possible outcomes is called the sample space. We
denote the sample space S.
o An event is a collection of outcomes from the sample space. To find
the probability of an event, add up the probabilities of all the
outcomes in the event.
 Rules of Probability
o The probability P(E) for any event E is always between 0 and 1,
inclusive. That is 0 ≤ P(E) ≤ 1.
o Law of Total Probability: for any experiment, the sum of all the
outcome probabilities in the sample space must equal 1.
 Classical Method of Assigning Probabilities
o Let N(E) and N(S) denote the number of outcomes in event E and the
sample space S, respectively. If the experiment has equally likely
outcomes, then the probability of event E is then
( )
( )
( )
Example 5.1
Q1. Find the probability of drawing an ace when drawing single card at random from a deck
of cards.
A1. There are 52 outcomes in the sample space, so N(S) = 52. Let E be the event that an
ace is drawn. Even E consists of the four aces {A♠, A♣, A♥, A♦}, so N(E) = 4.
Therefore, the probability of drawing an ace is 𝑷(𝑬)
𝑵(𝑬)
𝟒
𝟏
𝑵(𝒔)
𝟓𝟐
𝟏𝟑
2
 Relative Frequency Method of Assigning Probabilities
o The probability of event E is approximately equal to the relative
frequency of event E. that is,
( )
The relative frequency method is also known as the empirical method.
Example 5.2
Q1. This table contains the employment type for a sample of 1000 employed citizens of
Fairfax County, Virginia. Use the data to generate the relative frequencies and use the
frequencies to estimate the probabilities for each employment type.
Employment type
Private company
Federal government
Self-employed
Private nonprofit
Local government
State government
Other
Count
597
141
97
92
59
12
2
A1.
Employment type
Private company
Federal government
Self-employed
Private nonprofit
Local government
State government
Other
Probability
597/1000 = 0.597
141/1000 = 0.141
97/1000 = 0.097
92/1000 = 0.092
59/1000 = 0.059
12/1000 = 0.012
2/1000 = 0.002
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 Probabilities for Complements
o For any event A and its complement Ac, P(A) + P(Ac) = 1. Applying a
touch of algebra gives the following:
 P(A) = 1 – P(Ac)
 P(Ac)= 1 – P(A)
 Union and Intersection of Events
o The union of two events A and B is the event representing all the
outcomes that belong to A or B or both. The union of A and B is
denoted as A B and is associated with “or.”
o The intersection of two events A and B is the event representing all
the outcomes that belong to both A and B. the intersection of A and B
is denoted as A B and is associated with “and.”
 Addition Rule
(
)
( )
( )
(
)
Example 5.3
Q1. Suppose you pay $1 to play the following game. You choose one card at random from a deck of
52 cards, and you will win $3 if the card is either an ace or a heart. Find the probability of winning
this game.
A1. There are 4 aces in a deck of 52 cards, so by the classical method, P(A) = 4/52. There are 13
hearts in a deck of 52 cards, so P(H) = 13/52. We know that A H represents the ace of hearts.
P(Ace of hearts) = P(A H) = 1/52.
𝑷(𝑨
𝑯)
𝑷(𝑨)
𝑷(𝑯)
𝑷(𝑨
𝑯)
𝟒
𝟓𝟐
𝟏𝟑
𝟓𝟐
𝟏
𝟓𝟐
𝟒
𝟏𝟑
4
 Addition Rule for Mutually Exclusive Events
o If A and B are mutually exclusive events,
(
)
( )
( )
Example 5.4
Physical Appearance
Gender
Very attractive
Attractive
Average
Prefer not to
answer
Total
Female
Male
Total
3113
1415
4528
16,181
12,454
28,635
6,093
7,274
13,367
3478
2809
6287
28,865
23,952
52,817
Q1. Using this information, find the probability that a randomly chosen online dater has the following
characteristics.
a.
Is female.
There are a total of N(S) = 52,817 online daters in the entire data set. Of these, 28,865 are female,
denoted as event F. Therefore,
𝑷(𝑭)
b.
𝑷(𝑭𝒆𝒎𝒂𝒍𝒆)
𝑵(𝑭𝒆𝒎𝒂𝒍𝒆)
𝑵(𝑺)
𝑵(𝑭)
𝑵(𝑺)
𝟐𝟖, 𝟖𝟔𝟓
𝟓𝟐, 𝟖𝟏𝟕
𝟎. 𝟓𝟒𝟔𝟓
Self-reported as attractive
There are 28,635 people, who self-reported their physical appearance as attractive, denoted as event
A. Therefore,
𝑷(𝑨)
c.
𝑷(𝐒𝐞𝐥𝐟
𝑵(𝐒𝐞𝐥𝐟
𝐫𝐞𝐩𝐨𝐫𝐭𝐞𝐝 𝐚𝐭𝐫𝐫𝐚𝐜𝐭𝐢𝐯𝐞)
𝑵(𝑺)
𝑵(𝑨)
𝑵(𝑺)
𝟐𝟖, 𝟔𝟑𝟓
𝟓𝟐, 𝟖𝟏𝟕
𝑵(𝑭 𝑨)
𝑵(𝑺)
𝟏𝟔, 𝟏𝟖𝟏
𝟓𝟐, 𝟖𝟏𝟕
Is a female who self-reported as attractive
𝑷(𝑭𝒂𝒏𝒅𝑨)
d.
𝐫𝐞𝐩𝐨𝐫𝐭𝐞𝐝 𝐚𝐭𝐭𝐫𝐚𝐜𝐭𝐢𝐯𝐞)
𝟎. 𝟓𝟒𝟐𝟐
𝑷(𝑭
𝑨) 𝑷(𝐅𝐞𝐦𝐚𝐥𝐞 𝐚𝐧𝐝 𝐬𝐞𝐥𝐟
𝟎. 𝟑𝟎𝟔𝟒
𝐫𝐞𝐩𝐨𝐫𝐭𝐞𝐝 𝐚𝐭𝐭𝐫𝐚𝐜𝐭𝐢𝐯𝐞)
Is a female or self-reported as attractive
Here we seek P(F or A) = P(F
𝑷(𝑭
𝑨)
𝑷(𝑭)
A). By the Addition Rule,
𝑷(𝑨)
𝑷(𝑭
𝑨)
𝟎. 𝟓𝟒𝟔𝟓
𝟎. 𝟓𝟒𝟐𝟐
𝟎. 𝟑𝟎𝟔𝟒
𝟎. 𝟕𝟖𝟐𝟑
Q2. Find the probability that a randomly chosen online dater self-reported as either attractive or very attractive.
A1.
𝑵(𝑨)
𝟐𝟖,𝟔𝟑𝟓
𝑵(𝑺)
𝟓𝟐,𝟖𝟏𝟕
𝑷(𝑨)
𝑷(𝐬𝐞𝐥𝐟
𝐫𝐞𝐩𝐨𝐫𝐭𝐞𝐝 𝐚𝐭𝐫𝐫𝐚𝐜𝐭𝐢𝐯𝐞)
𝑷(𝑽)
𝑷(𝐬𝐞𝐥𝐟
𝐫𝐞𝐩𝐨𝐫𝐭𝐞𝐝 𝐯𝐞𝐫𝐲 𝐚𝐭𝐫𝐫𝐚𝐜𝐭𝐢𝐯𝐞)
𝟎. 𝟓𝟒𝟐𝟐
𝑵(𝑽)
𝟒𝟓𝟐𝟖
𝑵(𝑺)
𝟓𝟐,𝟖𝟏𝟕
𝟎. 𝟎𝟖𝟓𝟕𝟑
Since no online daters self-reported as both attractive and very attractive, the two groups are mutually
exclusive. Thus, by the Addition Rule for Mutually Exclusive Events,
𝑷(𝑨
𝑽)
𝑷(𝑨)
𝑷(𝑽)
𝟎. 𝟓𝟒𝟐𝟐
𝟎. 𝟎𝟖𝟓𝟕𝟑
𝟎. 𝟔𝟐𝟕𝟗𝟑
5
 Factorial Symbol n!
For any integer n ≥ 0, the factorial symbol n! is defined as follows:
o 0! = 1
o 1! = 1
o n! = n(n – 1)(n – 2) ... 3 * 2 * 1
 Permutations
A permutation is an arrangement of items, such that
o r items are chosen at a time from n distinct items.
o repetition of items is not allowed.
o the order of the items is important.
The number of permutations of n items chosen r at a time is denoted as nPr
and give by the formula:
nPr
(
)
Example 5.5
Q1. “Secret Santa” refers to a method whereby each member of group anonymously buys a holiday
gift for another member of the group. Each person is secretly assigned to buy a gift for another
randomly chosen person in the group. Suppose Jessica, Laverne, Samantha, and Luisa share a dorm
suite and would like to do Secret Santa this holiday season
a. Verify that in this instance one woman purchasing a gift for another woman represents a
permutation.



There are n = 4 women, and r = 2 two people are associated with each gift, the
give and the receivers.
Each person can buy only one gift, so repetition is not allowed.
Finally, there is a difference between Jessica buying for Laverne and Laverne
buying for Jessica. Thus, order is important, and thus, buying a gift represents a
permutation.
b. Calculate how many possible different permutations of gift buying there are for the four
women.
nPr
= 4P2
𝒏
𝟒
𝟒∗𝟑∗𝟐∗𝟏
(𝒏 𝒓)
(𝟒 𝟐)
𝟐∗𝟏
𝟏𝟐
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 Combinations
A combination is an arrangement of items in which
o r items are chosen from n distinct items.
o repetition of items is not allowed.
o the order of the items is not important.
The number of combinations of r items chosen from n different items is
denoted as
nCr
(
)
Example 5.6
Q1. “We return to the intramural singles tennis league at the local college. There are five players,
Ryan, Megan, Nicole, Justin, and Kyle. Each player must play each other once.
a. Confirm that a match between two players represents a combination



There are r = 2 players chosen from n = 5 players.
Each player plays each other player once, so repetition is not allowed.
There is no difference between {Ryan, Megan} and {Megan, Ryan}, so order is not
important.
b. How many matches will be held?
nCr
= 5C2
𝒏
𝟓
𝟓∗𝟒∗𝟑∗𝟐∗𝟏
𝒓 (𝒏 𝒓)
𝟐 (𝟓 𝟐)
𝟐∗𝟏(𝟑∗𝟐∗𝟏)
𝟏𝟎
{Ryan, Megan}{Ryan, Nicole}{Ryan, Justin}{Ryan, Kyle}
{Megan, Nicole}{Megan, Justin}{Megan, Kyle}
{Nicole, Justin}{Nicole, Kyle}
{Justin, Kyle}