Part C to G (Solution Required)

Texas holdem or, more simply, holdem, is now considered the most popular poker game. The Texas State Legislature officially recognizes Robstown, Texas, as the games birthplace and dates the game back to the early 1900s. Three reasons for the current popularity of Texas holdem can be attributed to (1) the emergence of Internet poker sites, (2) the hole cam (a camera that allows people watching television to see the hole cards of the players), and (3) Tennessee accountant and then-amateur poker-player Chris Moneymakers first-place win of $2.5 million in the 2003 World Series of Poker after winning his seat to the tournament through a $39 PokerStars satellite tournament. Following are the details of Texas holdem. Each player is dealt two cards face down, called hole cards, and then there is a betting round. Next, three cards are dealt face up in the center of the table. These three cards are termed the flop and are community cards, meaning that they can be used by all the players; again there is a betting round. Next, an additional community card is dealt face up, called the turn, and once again there is a betting round. Finally, a fifth community card is dealt face up, called the river, and then there is a final betting round. A player can use any five cards from the seven cards consisting of his two hole cards and the five community cards to constitute his or her hand. The player with the best hand (using the same handranking as in five card draw) wins the pot, that is, all the money that has been bet on the hand. There is one other way that a player can win the pot. Namely, if during any one of the four betting rounds all players but one have folded (i.e., thrown their hole cards face down in the center of the table), then the remaining player is awarded the pot. The best possible starting hand (hole cards) is two aces. What are the chances of being dealt those hole cards? Here we examine some of the simplest probabilities associated with the game. Recall that, to begin, each player is dealt 2 cards face down, called hole cards, from an ordinary deck of 52 playing cards. The best possible starting hand is two aces, referred to as pocket aces. a. The probability that you are dealt pocket aces is 1/221, or 0.00452 to three significant digits. If you studied either conditional probability and the multiplication rule or counting rule, verify that probability. b. Using the result from part (a), obtain the probability that you are dealt pocket kings. c. Using the result from part (a) and your analysis in part (b), find the probability that you are dealt a pocket pair, that is, two cards of the same denomination. Next recall that, after receiving your hole cards, there is a betting round. Subsequently, 3 cards, called the flop, are dealt face up in the center of the table. To do the remaining problems, you need to have studied either conditional probability and the multiplication rule or Section counting rule. Assuming that you are dealt a pocket pair, determine the probability that the flop d. contains at least 1 card of your denomination. (Hint: Complementation rule.) e. gives you trips, that is, contains exactly 1 card of your denomination and 2 other unpaired cards. f. gives you quads, that is, contains 2 cards of your denomination. g. gives you a boat, that is, contains 1 card of your denomination and 2 cards of another denomination.