HMMT 二月 2006 · 冲刺赛 · 第 34 题

HMMT February 2006 — Guts Round — Problem 34

[12] In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of “points” based on the formula 4 × (# A’s) + 3 × (# K’s) + 2 × (# Q’s) + 1 × (# J’s) . Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.

解析

In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of “points” based on the formula 4 × (# A’s) + 3 × (# K’s) + 2 × (# Q’s) + 1 × (# J’s) . Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher. 197 Answer: 1820 Solution: Obviously, we can ignore the cards lower than J. Simply enumerate the ways to get at least 13 points: AAAA (1), AAAK (16), AAAQ (16), AAAJ (16), AAKK (36), AAKQ (96), AKKK (16). The numbers in parentheses represent the number of ways to choose the suits, given the choices for the values. We see that there are a total of 1 + 16 + 16 + 16 + 36 + 96 + 16 = 197 ways to get at least 13. There are ( ) 16 a total of = 1820 possible ways to choose 4 cards from the 16 total A’s, K’s, Q’s, 4 and J’s. Hence the answer is 197 / 1820. 11