UTBM probability and statistics 2006 tc

SQ 28 Mid-term exam thMonday April 24 2006 For this test, you may use an electronic calculator and the Tables of Statistics. Length 2 hours. We consider a deck of 52 cards with 13 cards of each colour : Spades “, Hearts ', Diamonds ¤ and Clubs §. In each colour, there are 13 cards (values), in descending order, Ace = A, King = K, Queen = Q, Jack = J, 10, .., 3 and 2. These 52 cards are randomly dealt to four players sitting around a table, on the cardinal points : clockwise N(orth) E(ast), S(outh) and W(est). We call hand a set of 13 cards (order does not matter) out of the 52 cards. We call deal any sharing between 4 hands for 4 players. Two deals are therefore different if, and only if, at least one player has not got exactly the same hand. I. Q 1 – Counting 1–1 : What is the number of hands a player can get ? 1-2 : What is the number of deals four players can get ? 1-3 : What is the number of deals in which every player gets 13 cards of the same colour ? 1-4 : What is the number of deals in which a player gets 13 cards with 8 cards of one colour and 5 cards of another one ? -5 1-5 : What is the probability of this event (previous question). (answer = 3,13 . 10 ). Let T be the number of deals a player plays until he gets such a hand. What is the law of T ? 1-6 : Assuming that this player plays a tournament of 22 deals, 6 days a week an during a year, how long, on average, must he wait until he gets once more ...