10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCESTAT1301 Probability & Statistics IAssignment 1Due Date: September 24, 2010 (Hand in your solutions for Questions 3, 6, 15, 22, 23, 29, 30, 32) 1. A total of 8 different gifts are to be distributed among 12 children. How many distinct results are possible if no child is to receive more than one gift? 2. From a group of 9 women and 5 men a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if (a) 2 of the men refuse to serve together; (b) 2 of the women refuse to serve together; (c) 1 man and 1 woman refuse to serve together? 3. In how many ways can 10 people be seated in a row if (a) there are no restrictions on the seating arrangement; (b) persons A and B must sit next to each other; (c) there are 5 men and 5 women and no 2 men or 2 women can sit next to each other; (d) there are 6 men and they must sit next to each other; (e) there are 5 married couples and each couple must sit together? 4. Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to selected students from a class of 30. How many different outcomes are possible if (a) a student can receive any number of awards; (b) each student can receive at most 1 award? 5. (Poker) If it is assumed that all poker hands are equally likely, what is the probability of being dealt ⎟⎟⎠⎞⎜⎜⎝⎛552(a) one pair? (This occurs when the cards have denominations a, a, b, c, dwhere a, b, c, and dare all distinct.) (b) two pairs? (This occurs when the cards have denominations a, a, b, b, c, where a, b, and c are all distinct.) (c) three of a kind? (This occurs when the cards have denominations a, a, a, b, c.) (d) a straight? (Five consecutive cards of any suits which are not all the same, including 10, J, Q, K, A.) (e) a flush? (Five cards of the same suit and not all in consecutive order.) (f) full house? (This occurs when the cards have denominations a, a, a, b, b.) (g) four of a kind? (This occurs when the cards have denominations a, a, a, a, b.) (h)a straight-flush? (Five consecutive cards of the same suit, including 10, J, Q, K, A.) 6. From a group of 5 lawyers, 7 accountants, and 9 doctors, a committee of three is selected at random. What is the probability that the committee has more lawyers than doctors? P. 1
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