Question

Question 2 [20 marks A casino is analysing a poker game where 3 cards are dealt to the table by the dealer and two to a player, from a standard deck of 52 cards. Suppose 2 of the 3 cards on the table are Hearts and the player is holding 2 Hearts in his hand. The dealer's next action is to deal two more cards from the deck to the table. The player is interested in getting a flush which means he needs at least one more Heart to appear. He simulates some possibilities that the dealer might present him: 1] 'Club' 'Spade', [2] 'Spade' 'Club', [3] 'Heart' 'Club', [4] 'Spade' Diamond', [5 'Diamond' Heart', [6 'Club' 'Spade', [7] 'Heart' 'Diamond (a) (2 marks) Estimate the probability that at least one simulation. Heart is dealt from the (b) (5 marks) Let Y be the number of Hearts to appear among the last two cards to be dealt to the table. Write down the distribution that models Y and all of its parameters Note: Use the distribution determined in part (b) to answer the following questions. (c) (5 marks) What is the probability that one of the next two cards is a Heart? (d) (5 marks) What is the probability that neither one of the next two cards is a Heart? (e) (3 marks) Calculate the probability of receiving at least one tribution for Y. Compare this probability to the answer in part (a) and give Heart' with the dis- one reason why the answers are different.

Answer 1

(a) only 3 pos Observe that in the sample space of the simulation, there are sibilities of getting at least a heart. Hence the probability of getting atleast a heart is 3/7 (b) Observe that there are 4 hearts total in the deck and the hands of the player, and there is a getting y hearts is (y2) total of 5 cards dealt already. Hence the probability of = P(Y y) 13 4 (52 - 5) - (13 - 4) 2 y 52 - 5 2 47 9 2 y (C) 47 2 Hence Y Hypergeometric(47,9,2) (c) The probability that one of the two cards is hearts is = P(Y = 1) 38 20.32 47 2 (d) The probability that neither of the next two cards is Hearts is = P(Y = 0) 38 2 20.65 47