Question

A pinochle deck consists of two copies each of the 9, 10, J, Q, K and A cards of all 4 suits, for a total of 48 cards. You draw 5 cards at random from a pinochle deck

a. what is the probability of drawing a flush ( all 5 cards are the same suit)?

b. What is the probability of drawing a full house ( 3 cards of one value, 2 cards of a different value)?

Answer 1

Given the deck has 48 cards, with 24 cards repeated twice

Therefore, the number of ways in which you can draw 5 cards, this can be done in 3 ways

1) all cards are an identical number of ways = 24c5 = 42504

2) one pair of identical cards and the remaining 3 are different = 24c1 * 23c3 = 42504

3) two pairs are identical = 24c2 * 22c1 = 6072

Therefore the total number of ways = 42504+42504+6072 = 91080

a) probability drawing 5 cards from the same suit

number of ways of drawing = 4c1

therefore, probability = 4/91080

b) probability of drawing 3 cards of one value and 2 cards of different values

number of ways in which 3 cards of one value is drawn = 5c1 * 4c3 + 5c1* 4c1*3c1 (this includes both with repetition and without repetition) = 80

number of ways two cards of different values can be drawn = 4c2 = 6

Therefore, total number of ways =80*6 = 480

probability = 480/91080