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Thinking
Mathematically
Counting Methods and Probability
11.4 Fundamentals of Probability
Computing Theoretical
Probability
If an event E has n(E) equally-likely outcomes
and its sample space S has n(S) equally-likely
outcomes, the theoretical probability of event
E denoted by P(E), is
num berof outcom esin event E
P( E ) 
num berof outcom esin sam plespace S
Example Probability and a Deck of 52
Cards
You are dealt one card from a standard 52card deck. Find the probability of being
dealt a King.
Solution
Because there are 52 cards, the total number
of possible ways of being dealt a single card
is 52. We use 52, the total number of
possible outcomes, as the number in the
denominator. Because there are 4 kings in
the deck, the event of being dealt a king can
occur 4 ways.
P(king) = 4/52 = 1/13
More Examples
Exercise 11.4 #5, 21, 29, 37
A die is rolled, what is the probability of rolling a number
less than 3?
A fair coin is tossed two times, what is the probability of
getting two heads?
Arbitrarily selecting a family with three children, what is
the probability that the family has exactly two male
children?
A single die is rolled twice, what is the probability of
rolling two numbers such that their sum is 5?
Thinking
Mathematically
Counting Methods and Probability
11.5 Probability with the Fundamental Counting
Principle, Permutations, and Combinations
Example: Permutations
Exercise 11.5 #3
Six stand-up comics, A, B, C, D, E, and F are to perform
on a single evening. The order of performance is
determined randomly. Find the probability that
a. Comic E will perform first
b. Comic C will perform fifth and comic B will perform last.
c. The comedians will perform in the following order D, E, C,
A, B, F.
d. Comic A or comic B will perform first.
Example: Combinations
Exercise 11.5 # 5, #17
A group consists of four men and five women. Three people
are selected to attend a conference.
a.
b.
c.
In how many ways can three people be selected from this group
of nine?
In how many ways can three women be selected from the five
women?
Find the probability that the selected group will consist of all
women.
If you are dealt 4 cards from a shuffled deck of 52, find that
probability of getting two queens and two kings.
Thinking
Mathematically
Counting Methods and Probability
11.6 Events Involving Not and Or
The Probability of an Event Not
Occurring
The probability that an event E will not occur
is equal to 1 minus the probability that it
will occur.
P(not E) = 1 - P(E)
Exercise 11.6 #1
If you are dealt one card from a deck of 52, what is the
probability that the card is not an ace.
Mutually Exclusive Events
If it is impossible for events A and B to occur
simultaneously, the events are said to be
mutually exclusive.
If A and B are mutually exclusive events, then
P(A or B) = P(A) + P(B).
Exercise 11.6 #17
One card is randomly selected from a deck of 52. What is the
probability that the card is either a 2 or a 3?
Or Probabilities with Events That Are
Not Mutually Exclusive
If A and B are not mutually exclusive events,
then
P(A or B) = P(A) + P(B) - P(A and B)
Exercise 11.6 #33
You spin a spinner which has 8 equally likely outcomes,
numbered 1 through 8. What is the probability that
your spin will be either an even number or a number
greater than 5?
Thinking
Mathematically
Counting Methods and Probability
11.7 Events Involving And; Conditional Probability
Independent Events
Two events are independent events if the
occurrence of either of them has no effect
on the probability of the other.
If A and B are independent events, then
P(A and B) = P(A)•P(B)
Compare to P(A or B)
Example: And with Independent
Events
Exercises 11.7 #3, 13, 19
• A spinner has 3 red regions, 2 green regions, and 1
yellow regions. Each of the 6 regions are equally likely.
If the spinner is spun twice, what is the probability of
two successive yellow spins?
• A single die is rolled twice. What is the probability of
rolling an even number on the first roll and a number
greater than 2 the second time.
• A card is drawn from a deck of 52. The card is
replaced, the deck shuffled and a second card is drawn.
What is the probability of drawing a red card each time.
Dependent Events
Two events are dependent events if the
occurrence of one of them has an effect on
the probability of the other.
And Probabilities with Dependent
Events
If A and B are dependent events, then
P(A and B) = P(A)•P(B given that A has occurred)
The conditional probability of B, given A,
written P(B|A), is the probability that event B
occurs computed on the assumption that event
A occurs.
Example: And with Dependent
Events
Exercise 11.7 #43
An ice chest contains six cans of apple juice, eight cans of
grape juice, four cans of orange juice, and two cans of
mango juice. If you randomly choose three cans in
succession, what is the probability of choosing three
cans of apple juice?
Thinking
Mathematically
Chapter 11
Counting Methods and Probability