Section 7.5 Conditional Probability and Independent Events
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1 Section 75 Conditional Probability and Independent Events Conditional Probability of an Event If A and B are events in an experiment and P (A) 6= 0,thentheconditionalprobabilitythattheevent B will occur given that the event A has already occurred is P (B A) = P (A \ B) P (A) 1 A pair of fair 6sided dice is rolled What is the probability that a 2 is rolled if it is known that the sum of the numbers landing uppermost is less than or equal to 7? (Give answers as an exact fraction) 2 Acompanysurveyed1000peopleontheirageandthenumberofjeanspurchasedannuallyThe results of the poll are shown in the table orMore Total Ūnder o ver T otal A person is selected at random Use the table to answer these questions Round your answers to three decimal places (a) What is the probability that the person, who is over 18, purchases 2 pairs of jeans annually?
2 (b) What is the probability that a person who purchased less than 3 pairs of jeans each year will be in the age group 1218? Product Rule P (A \ B) =P (A) P (B A) 3 From the tree diagram find the following (a) P (A \ E) (b) P (A) (c) P (A E) 2 Fall 2017, Maya Johnson
3 4 In a survey of 1000 eligible voters selected at random, it was found that 200 had a college degree Additionally, it was found that 70% of those who had a college degree voted in the last presidential election, whereas 45% of the people who did not have a college degree voted in the last presidential election Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random will have the following characteristics (Round answers to three decimal places) (a) The voter had a college degree and voted in the last presidential election (b) The voter did not have a college degree and did not vote in the last presidential election (c) The voter voted in the last presidential election (d) The voter did not vote in the last presidential µ 3 Fall 2017, Maya Johnson
4 5 Two machines turn out all the products in a factory, with the first machine producing 75% of the product and the second 25% The first machine produces defective products 5% of the time and the second machine 7% of the time (a) What is the probability that a defective part is produced at this factory given that it was made on the first machine? (b) What is the probability that a defective part is produced at this factory? $0550 Independent Events If A and B are independent events, then P (A B) =P (A) and P (B A) =P (B) Test for the Independence of Two Events Two events A and B are independent if and only if P (A \ B) =P (A) P (B) 4 Fall 2017, Maya Johnson
5 6 The personnel department of Franklin National Life Insurance compiled the accompanying data regarding the income and education of its employees Income 60,000 or Below Income Above 60,000 Noncollege Graduate College Graduate Let A be the event that a randomly chosen employee has a college degree, and let B be the event that the chosen employee s income is more than $60, 000 (a) Find each of the following probabilities (Round answers to four decimal places) P (A) P (B) P (A \ B) P (B A) P (B A c ) (b) Are the events A and B independent events? 7 Suppose A and B are two events of a sample space S where P (A) =028, P (B) =024, and P (A [ B) =042 (a) What is P (A \ B)? (b) Are A and B independent events? 5 Fall 2017, Maya Johnson
6 03 8 An experiment consists of two independent trials The outcomes of the first trial are A, B, and C, with probabilities of occurring equal to 02, 02, and 06, respectively The outcomes of the second trial are E and F,withprobabilitiesofoccurringequalto03and07 Draw a tree diagram representing this experiment Use this tree diagram to find the probabilities below (a) P (B) (b) P (F B) (c) P (B \ F ) (d) P (F ) (e) Does P (B \ F )=P (B) P (F ) (f) Are B and F independent events? 9 Dystopia county has three bridges In the next year, the Elder bridge has an 8% chance of collapse, the Younger bridge has a 3% chance of collapse, and the Ancient bridge has a 19% chance of collapse What is the probability that exactly one of these bridges will collapse in the next year? (Round answer to four decimal places) 1 9) ( l 08 X )( l 1 9) t ( l 08 ) ( l 03 ) ( Fall 2017, Maya Johnson
7 10 If A and B are independent events, P (A) =035, and P (B) =055, find the probabilities below (Enter answers to four decimal places) (a) P (A \ B) (b) P (A [ B) (c) P (A B) (d) P (A c [ B c ) 7 Fall 2017, Maya Johnson
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