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    Simple Probability - Bayes Theorem and Venn Diagrams

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    1. In each one of the following cases, indicate classic probability is used, empirical (relative occurrence frequency) or subjective.
    a. A basketball player fails 30 of 50 distances. The probability that he fails the following shot is .60.
    b. A committee of seven students forms to study the environmental questions. What is the possibility that one of them is chosen like spokesman?
    c. You buy one of the 4 million tickets of Lottery. What is the probability that you win the prize of 1 million?
    d. The probability of a tremor in the north of California in next the 10 years is of 80, according to Mr. Ritcher Osmond, specialist in seismology.

    2. Out of 200 students in the business administration department, 50 are registered for Basic Accounting, 60 are registered for Economics and other 40 are taking both courses.
    a. Draw the Venn diagram.
    b. Determine the probability that a chosen student at random is in both courses.

    3. A professor has 30 students in two groups of accounting, Group A and Group B. The group A has 20 students, of which 5 will take the test for CPA. From group B, 4 will take the test for CPA. The other students will not take the test.
    a. Draw a probability tree representing the described situation.
    b. Determine the probability that a chosen student at random is going to take the test for CPA.

    4. Suppose that two events A and B are mutually exclusive. Determine the probability that they happen jointly.

    5. The manager of Puerto Rican Airlines is worried about the possibility of strikes in the company. He knows that the probability that their pilots go on strike is 0.75 and the probability that the drivers go on strike is 0.65. In addition, he knows that if the conductors go on strike, the pilots have a probability of .90 also of going on strike.
    a. Determine the probability that both groups go on strike.
    b. Determine the probability that the conductors go on strike if the pilots go on strike.

    6. The probability that a publicity campaign increases sales is .80. The probability that the cost of developing that campaign stays within the budget is .40. Suppose that the events are independent, determine the probability that the cost stays within budget.

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    Solution Preview

    Problem 1
    Classical probability is based on calculation of the likelihood without observation. Empirical probability is derived based on previous observation. Subjective probability is derived based on non-quantitative knowledge, such as a gut feeling.

    Thus a) is empirical, b) is classical, since you can calculate that one out of seven will be chosen, c) is classical, and d) is subjective.

    Problem 2
    A Venn diagram summarizes the number of students registered in each course and in both courses. See the link below for more details on Venn diagrams:

    See the Venn ...

    Solution Summary

    Problem 5
    a) Using Bayes' Theorem, we know that the probability of both groups striking is equal to P(pilots strike | conductors strike) * P(conductors strike) = 0.9 * 0.65 = 0.585
    b) Using Bayes' Theorem again, we know that P(conductors strike | pilots strike) = P(conductors strike AND pilots strike)/P(pilots strike) = 0.585/0.75 = 0.78