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8 张牌中的梅花数

8 Clubs

专题
Statistics / 统计
难度
L3

题目详情

从一副标准 52 张扑克牌中发出 8 张。求这 8 张牌中梅花数量的方差。

8 cards are dealt out from a standard deck. Find the variance of the number of clubs in the 8 cards.

解析

设 XX 表示从一副标准 52 张牌中发出的 8 张牌里梅花的数量。

XX 服从参数为下列数值的超几何分布： N=52,K=13,n=8N = 52,\quad K = 13,\quad n = 8

超几何随机变量的方差为 Var⁡(X)=nKN(1−KN)N−nN−1.\operatorname{Var}(X) = n \frac{K}{N}\left(1 - \frac{K}{N}\right)\frac{N-n}{N-1}.

代入数值： Var⁡(X)=8⋅1352⋅3952⋅4451.\operatorname{Var}(X) = 8 \cdot \frac{13}{52} \cdot \frac{39}{52} \cdot \frac{44}{51}.

化简： 1352=14,3952=34.\frac{13}{52} = \frac{1}{4}, \quad \frac{39}{52} = \frac{3}{4}.

因此， Var⁡(X)=8⋅14⋅34⋅4451=32⋅4451=6651=2217.\operatorname{Var}(X) = 8 \cdot \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{44}{51} = \frac{3}{2} \cdot \frac{44}{51} = \frac{66}{51} = \frac{22}{17}.

Var⁡(X)=2217\boxed{\operatorname{Var}(X) = \frac{22}{17}}

Original Explanation

Let XX be the number of clubs in 8 cards dealt from a standard 52-card deck.

XX has a hypergeometric distribution with parameters: N=52,K=13,n=8N = 52,\quad K = 13,\quad n = 8

The variance of a hypergeometric random variable is Var⁡(X)=nKN(1−KN)N−nN−1.\operatorname{Var}(X) = n \frac{K}{N}\left(1 - \frac{K}{N}\right)\frac{N-n}{N-1}.

Substitute the values: Var⁡(X)=8⋅1352⋅3952⋅4451.\operatorname{Var}(X) = 8 \cdot \frac{13}{52} \cdot \frac{39}{52} \cdot \frac{44}{51}.

Simplify: 1352=14,3952=34.\frac{13}{52} = \frac{1}{4}, \quad \frac{39}{52} = \frac{3}{4}.

Thus, Var⁡(X)=8⋅14⋅34⋅4451=32⋅4451=6651=2217.\operatorname{Var}(X) = 8 \cdot \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{44}{51} = \frac{3}{2} \cdot \frac{44}{51} = \frac{66}{51} = \frac{22}{17}.

Var⁡(X)=2217\boxed{\operatorname{Var}(X) = \frac{22}{17}}