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两个孩子：至少一个男孩 vs 看到一个男孩

Boys and Girls

专题
Probability / 概率
难度
L4

题目详情

A. 某公司为“至少有一个儿子”的双职工妈妈办晚宴。已知 Jackson 女士有两个孩子且至少有一个是男孩。问两个都是男孩的概率？

B. 你新同事 Parker 女士也有两个孩子。你在路上看到她带着其中一个孩子，而那个孩子是男孩。问两个都是男孩的概率？

A. A company is holding a dinner for working mothers with at least one son. Ms. Jackson, a mother of two children, is invited. Given that she has at least one boy, what is the probability that both children are boys?

B. Your new colleague, Ms. Parker, also has two children. You see her walking with one of them, and that child is a boy. What is the probability that both children are boys?

解析

设样本空间为 {(b,b),(b,g),(g,b),(g,g)}\{(b,b),(b,g),(g,b),(g,g)\}（老大/老二）。

  • A：已知“至少一个男孩”，排除 (g,g)(g,g)，剩 3 种等可能，因此概率 13\frac{1}{3}。
  • B：若“看到的那个孩子”可视作从两个孩子中随机观察到的一个，则在观察到男孩的条件下，两男概率为 12\frac{1}{2}。

Original Explanation

Let the sample space be Ω={(b,b),(b,g),(g,b),(g,g)}\Omega = \{(b,b), (b,g), (g,b), (g,g)\}, representing (older child, younger child) as boy (b) or girl (g).

  • Part A
    We know Ms. Jackson has at least one boy. Hence the relevant cases are {(b,b),(b,g),(g,b)}\{(b,b), (b,g), (g,b)\}. The probability both children are boys is
    P({(b,b)})P({(b,b),(b,g),(g,b)})=13.\frac{P(\{(b,b)\})}{P(\{(b,b), (b,g), (g,b)\})} = \frac{1}{3}.

  • Part B
    You observe one child is a boy. Assuming that observed boy is chosen randomly (and not necessarily the older one), the conditional probability that both children are boys becomes 12.\frac{1}{2}.