11.6 Conditional Probability In general, the probability of event E2 occurring, given that an event E1 has happened (or will happen; the time relationship does not matter), is called a conditional probability and is written P(E2|E1). Example 1: A single card is selected from a deck of cards. Determine the probability it is a club, given that it is black. Example 2: A letter is randomly selected from the letters of the English alphabet. Find the probability of selecting a vowel, given that the outcome is a letter that precedes h

Example 3: If one person from the 250 patients surveyed is selected at random, determine the probability that the person was satisfied with the results of the surgery. was satisfied with the results of the surgery, given that the person had knee surgery. was dissatisfied with the results of the surgery, given that the person had hip surgery. had heart surgery, given that the person was dissatisfied with the results of the surgery.

Example 4: Sally, a quality control inspector, is checking a sample of lightbulbs for defects. The following table summarizes her findings. If one of these light bulbs is selected at random, determine the probability that the lightbulb is Good. Good, given that it is 50 watts. Good, given that it is 100 watts. Good, given that it is 50 or 100 watts. Defective, given that it is not 50 watts.

11.7 Counting Principle and Permutations Example 1 (Review): A password used to gain access to a computer account is to consist of two lower case letters followed by four digits. Determine how many different passwords are possible if repetition of letters and digits is permitted. b) repetition of letters and digits is not permitted. c) the first letter must be a vowel (a, e, i, o, u) and the first digit cannot be a 0, and repetition of letters and digits is not permitted.

n! = n(n  1)(n  2)···(3)(2)(1) Permutation is an ordered arrangement of items that occurs when: No item is used more than once. The order of arrangement makes a difference. The product 7∙6∙5∙4∙3∙2∙1 is called 7 factorial and is written 7! If n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n down through 1. n! = n(n  1)(n  2)···(3)(2)(1) 0! (zero factorial), by definition, is 1. 0! = 1

Example 2: Evaluate Example 3: Consider the five letters a, b, c, d, e. In how many distinct ways can three letters be selected and arranged if repetition is not allowed?

The number of possible permutations of r items are taken from n items: Example 4 (redone): Consider the five letters a, b, c, d, e. In how many distinct ways can three letters be selected and arranged if repetition is not allowed? Example 5: You are among eight people forming a skiing club. Collectively, you decide to put each person’s name in a hat and to randomly select a president, a vice president, and a secretary. How many different arrangements or permutations of officers are possible? Example 6: You and 19 of your friends have decided to form a business. The group needs to choose three officers– a CEO, an operating manager, and a treasurer. In how many ways can those offices be filled?

The number of distinct permutations of n objects where n1 of the objects are identical, n2 of the objects are identical, …, nr of the objects are identical is found by the formula Example 7: In how many different ways can the letters of the word BANANAS be arranged? Example 8: In how many different ways can the letters of the word TALLAHASSEE be arranged? Example 9: In how many distinct ways can the letters of the word MISSISSIPPI be arranged?

11.8 Combinations A combination of items occurs when The items are selected from the same group. No item is used more than once. The order of items makes no difference. Note: Permutation problems involve situations in which order matters. Combination problems involve situations in which the order of items makes no difference.

Examples 1 – 2: Determine which involve permutations and which involve combinations. Example 1: Six students are running for student government president, vice-president and treasure. The student with the greatest number of votes becomes the president, the second highest vote-getter becomes vice-president, and the student who gets the third largest number of votes will be treasurer. How many different outcomes are possible? Example 2: Six people are on the board of supervisors for your neighborhood park. A three-person committee is needed to study the possibility of expanding the park. How many different committees could be formed from the six people?

Comparing Combinations and Permutations Given the letters: A,B,C,D: We can compare how many permutations and how many combinations are possible if we chose 3 letters at a time:

The number of possible combinations if r items are taken from n items is: Example 3: While visiting New York City, the Friedmans are interested in visiting 8 museums but have time to visit only 3. In how many ways can the Friedmans select 3 of the 8 museums to visit?

Example 4: Jan Funkhauser has 10 different cut flowers from which she will choose 6 to use in a floral arrangement. How many different ways can she do so? Example 5: How many three-person committees could be formed from 8 people?

Example 6: At the Royal Dynasty Chinese restaurant, dinner for eight people consists of 3 items from column A, 4 items from column B, and 3 items from column C. If columns A, B, and C have 5, 7, and 6 items, respectively, how many different dinner combinations are possible? Example 7: In December, 2009, the U.S Senate consisted of 60 Democrats and 40 Republicans. How many committees can be formed if each committee must have 3 Democrats and 2 Republicans?