Math115a Bayes’ Theorem #2 Name:________________________For each problem on the worksheet: Define the random variables used. Use the correct mathematical notation. Use a tree diagram, Venn diagram, table or other method to show all the probabilities given in the problem.1. There are 22 students in a math 115a class. 11 of these students took math 110 at the U of A. 15 of the students in the class are male. 8 of the male students took 110 at the U of A. Let M=be the event that a student is male, 110= the event that a student took 110 at the U of A. Make a Venn diagram representing this information. Calculate:a. P(M)b. P(MC)c. P(M110)d. P(M|110)2. For a fixed length of time, the probability of worker error on a certain production line is .1 , the probability that an accident will occur when there is a worker error is .3, and the probability that an accident will occur when there is no worker error is .2 . Let A represent the event of an accident, and let E represent the event of a worker error.a. P(E)b. P(EC)c. P(A)d. P(A|E)e. P(AC|E)f. P(A|EC)g. P(AC|EC)h. Find the probability of a worker error if there is an accident. i. Find the probability of a worker error if there is no accident.3. Your retail business is considering holding a sidewalk sale promotion next Saturday. Past experience indicates that the probability of a successful sale is 60%, if it does not rain. This drops to 30% if it does rain on Saturday. A phone call to the weather bureau finds an estimated probability of 20% for rain. Let R be the event that it rains on Saturday. Let S be the event that the sale on Saturday is a success. Answer the following in terms of this problem: (Use correct mathematical notation for each problem)a. What is the probability of rain? b. What is the probability of no rain?c. What is the probability that the sale is successful if it does not rain? d. What is the probability that the sale is unsuccessful if it does not rain?e. What is the probability that the sale is successful if it does rain?4. A Jar I contains 4 red balls and Jar II contains 3 red balls and 2 blue. The experiment is to chose a jarat random and from this jar select a ball and note the color of the ball. What is the probability that a blue ball is drawn given that the ball is from Jar II?5. All tractors made by a company are produced on one of three assembly lines, named Red, White, and Blue. The chances that a tractor will not start when it rolls off of a line are 6%, 11%, and 8% for lines Red, White, and Blue, respectively. 48% of the company’s tractors are made on the Red lineand 31% are made on the Blue line. What fraction of the company’s tractors do not start when they roll off of an assembly line?6. Thirty percent of the population has a certain disease. Of those that have the disease, 89% will test positive for the disease. Of those that do not have the disease, 5% will test positive. What is the probability that a person has the disease, given that they test positive for the disease?b. What is the probability that a person who has the disease will test positive?c. What is the probability that a person who is disease free will test negative?d. What is the probability that a person tests negative given that he does not the disease?7. A test attempts to recognize the presence of a certain disease. Records show that 10% of adults have a strong form of the disease, 18% have a mild form of the disease, and the rest have no form of the disease. A person with a strong form of the disease has a 15% chance of testing negative. A person with a mild form has a 10% chance of testing negative. A person who does not have the disease has a13% chance of testing positive, thereby falsely indicating that the person has the disease.a. If a person tests positive for the disease, what is the probability that he or she actually has the strong form of the disease? b. If a person tests positive for the disease, what is the probability that he or she actually has the mild form of the disease? c. If a person tests negative for the disease, what is the probability that he or she actually does not have either form of the
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