Download Ch 4

Ch5 Probability
Ch 5.4 Conditional Probability and the General Multiplication Rule
Objective A : Conditional Probability and the General Multiplication Rule
A1. Multiplication Rule for Dependent Events
If E and F are dependent events, then P( E and F )  P( E )  P( F | E ) .
The probability of E and F is the probability of event E occurring times the probability of event F
occurring, given the occurrence of event E .
Example 1: A box has 5 red balls and 2 white balls. If two balls are randomly selected (one after the
other), what is the probability that they both are red?
(a) With replacement
(b) Without replacement
Example 2: Three cards are drawn from a deck without replacement. Find the probability that all are
jacks.
A2. Conditional Probability
If E and F are any two events, then P( F E ) 
P( E and F ) N ( E and F )
.

P( E )
N (E)
The probability of event F occurring, given the occurrence of event E , is found by dividing the
probability of E and F by the probability of E .
Example 1: At a local Country Club, 65% of the members play bridge and swim, and 72% play bridge. If a
member is selected at random, find the probability that the member swims, given that the
member plays bridge.
B. Application
Example 1: Eighty students in a school cafeteria were asked if they favored a ban on smoking in the
cafeteria. The results of the survey are shown in the table.
Class
Favor
Oppose
No opinion
Freshman
15
27
8
Sophomore
23
5
2
If a student is selected at random, find these probabilities.
(a) The student is a freshman or favors the ban.
(b) Given that the student favors the ban, the student is a sophomore.
Example 2 : The local golf store sells an “onion bag” that contains 35 “experienced” golf balls. Suppose
that the bag contains 20 Titleists, 8 Maxflis and 7 Top-Flites.
(a) What is the probability that two randomly selected golf balls are both Titleists?
(b) What is the probability that the first ball selected is a Titleist and the second is a Maxfli?
(c) What is the probability that the first ball selected is a Maxfli and the second is a Titleist?
(d) What is the probability that one golf ball is a Titleist and the other is a Maxfli?
Ch 5.5 Counting Techniques
Objective A : Multiplication Rule of Counting
If Task 1 can be done in n ways and Task 2 can be done in m ways, Task 1 and Task 2 performed
together can be done in nm ways.
Example 1: Two dice are tossed. How many outcomes are in the sample space?
Example 2: A password consists of two letters followed by one digit. How many different
passwords can be created? (Note: Repetitions are allowed)
Objective B : Counting Techniques : Permutation and Combination
Permutation
The number of ways we can arrange n distinct objects, taking them r at a time, is
n!
Order Matters
n Pr 
(n  r )!
Combination
The number of distinct combinations of n distinct objects that can be formed, taking them r at
a time, is
n
Cr 
n!
r !(n  r )!
Example 1: Find (a) 5!
Order doesn’t matter
(b)
P
25 8
(c)
12
P4
(d)
25
C8
Example 2: An inspector must select 3 tests to perform in a certain order on a manufactured
part. He has a choice of 7 tests. How many ways can be performed 3 different
tests?
Example 3: If a person can select 3 presents from 10 presents, how many different
combinations are there?
Objective C: Using the Counting Techniques to Find Probabilities
After using the multiplication rule, combination, and permutation learned from this section to
count the number of outcomes for a sample space, N ( S ) , and the number of outcomes for an
N (E)
event, N ( E ) , we can calculate P ( E ) by the formula P( E ) 
N (S )
Example 1: A Social Security number is used to identify each resident of the United States
uniquely. The number is of the form xxx-xx-xxxx, where each x is a digit from 0 to 9.
(a) How many Social Security numbers can be formed?
(b) What is the probability of correctly guessing the Social Security number of the
President of the United States?
Example 2 : Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A
committee of seven senators is to be formed by selecting members of the Senate
randomly.
(a) What is the probability that the committee is composed of all Democrats?
(b) What is the probability that the committee is composed of all Republicans?
(c) What is the probability that the committee is composed of all three Democrats and
four Republicans?