Question: Refer to Example. For each poker holding below,

(1) find the number of five-card poker hands with that holding;

(2) find the probability that a randomly chosen set of five cards has that holding.

a. royal flush

b. straight flush

c. four of a kind

d. full house

e. flush

f. straight

g. three of a kind

h. one pair

i. neither a repeated denomination nor five of the same suit nor five adjacent denominations

Example: Poker Hand Problems

The game of poker is played with an ordinary deck of cards. Various five-card holdings are given special names, and certain holdings beat certain other holdings. The named holdings are listed from highest to lowest below.

Royal flush: 10, J, Q, K, A of the same suit Straight flush: five adjacent denominations of the same suit but not a royal flush-aces can be high or low, so A, 2, 3, 4, 5 of the same suit is a straight flush. Four of a kind: four cards of one denomination-the fifth card can be any other in the deck Full house: three cards of one denomination, two cards of another denomination Flush: five cards of the same suit but not a straight or a royal flush Straight: five cards of adjacent denominations but not all of the same suit-aces can be high or low Three of a kind: three cards of the same denomination and two other cards of different denominations Two pairs: two cards of one denomination, two cards of a second denomination, and a fifth card of a third denomination One pair: two cards of one denomination and three other cards all of different denominations No pairs: all cards of different denominations but not a straight, or straight flush, or flush, or royal flush

a. How many five-card poker hands contain two pairs?

b. If a five-card hand is dealt at random from an ordinary deck of cards, what is the probability that the hand contains two pairs?