Conditional Probability Honors

Conditional Probability  The probability of an event occurring, given that another event has already occurred  Denoted P(B | A) (read “probability of B, given A”)  (probability of event B occuring, given that event A has occurred)

 Two Way Frequency Table  A frequency table that contains data from two different categories 3

 The table shows students by gender and by type of school in You pick a student at random. What is P(female | graduate school)? 4

 The table shows students by gender and by type of school in You pick a student at random. What is P(female)? 5

 Americans recycle increasing amounts through municipal waste collection. The table shows the collection data for What is the probability that a sample of recycled waste is paper? 6

Example A recent survey of 400 instructors at a major university revealed the data shown in the following table. Based on the data, what are the probabilities of the following? a. An instructor received a good evaluation, given the instructor was tenured. b. An instructor received a poor evaluation, given the instructor was tenured. 7

Solutions Status Good Evaluations Poor Evaluation s Tenured72168 Nontenured8476 8

Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced.) Solution: Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens.

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 A utility company asked 50 of its customers whether they pay their bills online or by mail. What is the probability that a customer pays the bill online, given that the customer is a male? 11

Researchers asked people who exercise regularly whether they jog or walk. Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the probability that a male respondent jogs. Relate:P( male ) = 58% P( male and jogs ) = 20% Define:Let A = male. Let B = jogs. The probability that a male respondent jogs is about 34%. Write:P( A | B ) = P( A and B ) P( A ) = Substitute 0.2 for P(A and B) and 0.58 for P(A) Simplify

From That P(A and B) = P(A) * P(B | A) 13

 A school system compiled the following information from a survey it sent to people who were juniors 10 years earlier. ◦ 85% of the students graduated from high school ◦ Of the students who graduated from high school, 90% are happy with present job ◦ Of the students who did not graduate from high school, 60% are happy with their present job. ◦ What is the probability that a person from the junior class 10 years ago graduated from high school and is happy with his/her present job? 14

◦ 85% of the students graduated from high school ◦ Of the students who graduated from high school, 90% are happy with present job ◦ Of the students who did not graduate from high school, 60% are happy with their present job. ◦ What is the probability that a person from the junior class 10 years ago graduated from high school and is happy with his/her present job? 15

Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen. Solution: Because the first card is not replaced, the events are dependent.

1. Use the table to find each probability a. P(has diploma) b. P(has diploma and experience) c. P(has experience | has diploma) d. P(has no diploma | has experience) 2. Researchers asked shampoo users whether they apply shampoo directly to the head, or indirectly using a hand. What is the probability that a respondent applies shampoo directly to the head, given that the respondent is female? 3. Use the survey results a. Find the probability that a respondent has a pet, given the respondent has had a pet. b. Find the probability that a respondent has never had a pet given that the respondent does not have a pet now. 17