Probability and Poker Some mathematical exercises by Jeffrey rosenthal, Department of statistics, University of Toronto May1,2000 This exercise computes various probabilities related to the game of poker Recall that an ordinary 52-card deck of cards consists of 52 cards, of which 13 each are Clubs, Diamonds, Hearts, and Spades. Recall that an ordinary poker hand consists of 5 cards, chosen uniformly at random from an ordinary 52-card deck. Recall that a poker hand is a flush if all 5 cards are of the same suit, i.e. either all Clubs, or all Diamonds, or all Hearts, or all Spades 1. Compute the probability that a given ordinary poker hand is a fush 2. In some poker games, the cards are dealt out a few at a time, rather than all at once Suppose a player already has three cards, and they are all Clubs. The player will then be dealt two more cards, chosen uniformly at random from the remainder of the deck Compute the probability that they will end up with a flush. How does this probability compare with your answer to Question 1? 3. In some poker games(e.g. "Five Card Stud"), a player gets to see some of their is such that each player has been dealt three cards, of which two are "up caldo pponents cards(the "up cards"), before the hand is complete. Suppose a game Each player will later be dealt two more cards, chosen uniformly at random from the remainder of the deck. Suppose there are five players in the game. Suppose that one player(“ Player#1”) has all three cards Clubs (a)Suppose further that, of all the up cards of all the other four players, none of them are Clubs. Compute the probability that Player #1 will end up with a fush (b) Suppose instead that, of all the up cards of all the other four players, all of them are Clubs. Compute the probability that Player#l will end up with a flush (c) Suppose the situation(because of the betting so far)is such that Player #1 will fold (i.e, drop out of the game) unless they have at least a 3% chance of getting a flush. Compute the smallest number of Clubs among the up cards of all the other four players, such that Player #1 will fold (d)What conclusions can be drawn from this question, regarding an actual game of 4. In some poker games(e.g. "Seven Card Stud), players get more than five cards, and they then get to choose which five cards count as their final poker hand. In this case 1Probability and Poker Some mathematical exercises by Jeffrey Rosenthal, Department of Statistics, University of Toronto May 1, 2000. This exercise computes various probabilities related to the game of poker. Recall that an ordinary 52-card deck of cards consists of 52 cards, of which 13 each are Clubs, Diamonds, Hearts, and Spades. Recall that an ordinary poker hand consists of 5 cards, chosen uniformly at random from an ordinary 52-card deck. Recall that a poker hand is a flush if all 5 cards are of the same suit, i.e. either all Clubs, or all Diamonds, or all Hearts, or all Spades. 1. Compute the probability that a given ordinary poker hand is a flush. 2. In some poker games, the cards are dealt out a few at a time, rather than all at once. Suppose a player already has three cards, and they are all Clubs. The player will then be dealt two more cards, chosen uniformly at random from the remainder of the deck. Compute the probability that they will end up with a flush. How does this probability compare with your answer to Question 1? 3. In some poker games (e.g. “Five Card Stud”), a player gets to see some of their opponents cards (the “up cards”), before the hand is complete. Suppose a game is such that each player has been dealt three cards, of which two are “up cards”. Each player will later be dealt two more cards, chosen uniformly at random from the remainder of the deck. Suppose there are five players in the game. Suppose that one player (“Player #1”) has all three cards Clubs. (a) Suppose further that, of all the up cards of all the other four players, none of them are Clubs. Compute the probability that Player #1 will end up with a flush. (b) Suppose instead that, of all the up cards of all the other four players, all of them are Clubs. Compute the probability that Player #1 will end up with a flush. (c) Suppose the situation (because of the betting so far) is such that Player #1 will fold (i.e., drop out of the game) unless they have at least a 3% chance of getting a flush. Compute the smallest number of Clubs among the up cards of all the other four players, such that Player #1 will fold. (d) What conclusions can be drawn from this question, regarding an actual game of poker? 4. In some poker games (e.g. “Seven Card Stud”), players get more than five cards, and they then get to choose which five cards count as their final poker hand. In this case, 1