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The third exam for math 1320, focusing on probability theory and set theory. The exam includes problems on finding probability distributions, expressing events with dice rolls, analyzing sets, and computing theoretical probabilities. Students are required to show their work for full credit.
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Name:__________________________________________ ID number:______________ Instructions: Solve the following problems. Show all your work in order to get full credit. Problem 1. A die is weighted in such a way that each of 1, and 3 is twice as likely to come up as each of 2, 4, 5, and 6. Find the probability distribution. What is the probability of rolling less or equal than 4?
Problem 2. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 2; B: the numbers add to 3; C: at least one of the numbers is 1; and D: the numbers do not add to 11. Express the given event in symbols and say how many elements it contains.
Problem 4. The following table, which shows the profile, by Math SAT I scores, of admitted students at UCLA for the Fall 2004 semester 200 - 399 400 - 499 500 - 599 600 - 699 700 - 799 Total Admitted 7 212 1124 2882 5309 9534 Not Admitted 687 3512 8689 12230 5150 30268 Total Applicants 694 3724 9813 15112 10459 39802 Compute the theoretical probabilities of the following events:
Problem 5. If P ( A ∪ B ) =. 4, P ( A ∩ B ) =. 1, and P ( B )=0.2. Find P ( A ). Problem 6. If 12 businesspeople have a meeting and each pair exchanges business cards, how many business cards, total, get exchanged? Problem 7. A bag contains 5 red marbles, 3 green ones, 2 lavender one, 4 yellows, and 2 orange marbles. a. How many sets of five marbles include one of each color? b. How many sets of five marbles include at least two red ones?