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12.2.1: Bayes' Formula (Exercises)

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    SECTION 9.2 PROBLEM SET: BAYES' FORMULA

    1. Jar I contains five red and three white marbles, and Jar II contains four red and two white marbles. A jar is picked at random and a marble is drawn. Draw a tree diagram below, and find the following probabilities.
      1. P(marble is red)
      2. P(It came from Jar II | marble is white)
      3. P(Red | Jar I)
    1. In Mr. Symons' class, if a student does homework most days, the chance of passing the course is 90%. On the other hand, if a student does not do homework most days, the chance of passing the course is only 20%.
      H = event that the student did homework
      C = event that the student passed the course
      Mr. Symons claims that 80% of his students do homework on a regular basis. If a student is chosen at random from Mr. Symons' class, find the following probabilities.
      1. P(C)
      2. P(H|C)
      3. P(C|H)
    1. A city has 60% Democrats, and 40% Republicans. In the last mayoral election, 60% of the Democrats voted for their Democratic candidate while 95% of the Republicans voted for their candidate. Which party's mayor runs city hall?
    1. In a certain population of 48% men and 52% women, 56% of the men and 8% of the women are color-blind.
      1. What percent of the people are color-blind?
      2. If a person is found to be color-blind, what is the probability that the person is a male?
    1. A test for a certain disease gives a positive result 95% of the time if the person actually carries the disease. However, the test also gives a positive result 3% of the time when the individual is not carrying the disease. It is known that 10% of the population carries the disease. If a person tests positive, what is the probability that he or she has the disease?
    1. A person has two coins: a fair coin and a two-headed coin. A coin is selected at random, and tossed. If the coin shows a head, what is the probability that the coin is fair?
    1. A computer company buys its chips from three different manufacturers. Manufacturer I provides 60% of the chips and is known to produce 5% defective; Manufacturer II supplies 30% of the chips and makes 4% defective; while the rest are supplied by Manufacturer III with 3% defective chips. If a chip is chosen at random, find the following probabilities:
      1. P(the chip is defective)
      2. P(chip is from Manufacturer II | defective)
      3. P(defective |chip is from manufacturer III)
    1. Lincoln Union High School District is made up of three high schools: Monterey, Fremont, and Kennedy, with an enrollment of 500, 300, and 200, respectively. On a given day, the percentage of students absent at Monterey High School is 6%, at Fremont 4%, and at Kennedy 5%. If a student is chosen at random, find the probabilities below: Hint: Convert the enrollments into percentages.
      1. P(the student is absent)
      2. P(student is from Kennedy | student is absent)
      3. P(student is absent | student is from Fremont)

    9. At a retail store, 20% of customers use the store’s online app to assist them when shopping in the store ; 80% of store shoppers don’t use the app.

    Of those customers that use the online app while in the store, 50% are very satisfied with their purchases, 40% are moderately satisfied, and 10% are dissatisfied.

    Of those customers that do not use the online app while in the store, 30% are very satisfied with their purchases, 50% are moderately satisfied and 20% are dissatisfied.

    Indicate the events by the following:

    A = shopper uses the app in the store
    N = shopper does not use the app in the store
    V = very satisfied with purchase
    M = moderately satisfied
    D = dissatisfied

    a. Find P(A and D), the probability that a store customer uses the app and is dissatisfied

    b. Find P(A|D), the probability that a store customer uses the app if the customer is dissatisfied.

    10. A medical clinic uses a pregnancy test to confirm pregnancy in patients who suspect they are pregnant. Historically data has shown that overall, 70% of the women at this clinic who are given the pregnancy test are pregnant, but 30% are not.

    The test's manufacturer indicates that if a woman is pregnant, the test will be positive 92% of the time.

    But if a woman is not pregnant, the test will be positive only 2% of the time and will be negative 98% of the time.

    a. Find the probability that a woman at this clinic is pregnant and tests positive.

    b. Find the probability that a woman at this clinic is actually pregnant given that she tests positive.

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