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NO.PZ2018122801000006

问题如下：

In country X, the probability that a letter sent through the postal system reaches its destination is 2/3. Assume that each postal delivery is independent of every other postal delivery, and assume that if a wife receives a letter from her husband, she will certainly mail a response to her husband. Suppose a man in country X mails a letter to his wife (also in country X) through the postal system. If the man does not receive a response letter from his wife, what is the probability that his wife received his letter?

选项：

1/3

3/5

2/3

2/5

解释：

D is correct.

考点 $：$ Bayes' formula

解析 $：$

(Notices: if the wife did not receive the letter, she would not send response to the man.)

Note:

A: the wife received the letter; $\bar{A}$ : the wife did not receive the letter;

B: the man received a response letter from his wife; $\bar{B}$ : the man did not receive a response letter from his wife

The question "If the man does not receive a response letter from his wife, what is the probability that his wife received his letter?" is equal to figure out $P (A | \bar{B})$

考点：正态分布查表

$P (A | \bar{B}) = \frac{P (A \bar{B})}{P (\bar{B})} = \frac{P (\bar{B}) \times P (\bar{B} | A)}{P (\bar{B}) \times P (\bar{B} | A) + P (\bar{B}) \times P (\bar{B} | \bar{A})}$ 解析：题目说信可以成功寄到的概率是2/3，假设每次送信都是独立事件。同时假设妻子如果收到丈夫的信就一定会回复。问题问如果丈夫没有收到回信，那她的妻子收到丈夫的信的概率是多少？

首先丈夫没有收到回信的条件有两种情况：1.丈夫的寄信丢了，就是上面那个二叉树的下面一根，概率是1/3；2.丈夫的信寄到了，但是妻子的回信寄丢了（1/3），这种情况下概率为1/3*（2/3）=2/9

所以这两种情况下概率加和=1/3+2/9=5/9

根据贝叶斯公式，

如果丈夫没有收到回信，那她的妻子收到丈夫的信的概率=（2/9）/(5/9)=2/5

$= \frac{(2 / 3) \times (1 / 3)}{(2 / 3) \times (1 / 3) + (1 / 3) \times 1} = \frac{2}{5}$

请老师用中文详细解释一下，我考中文

题目：在一个国家收到寄信的可能性是2/3 （也就是说信寄丢的可能性是1/3）；然后假设夫妻两人都在这同一个国家（以上概率适用于两个人）并且如果丈夫给妻子寄信只要妻子收到了丈夫的信就一定会回信

问题：如果妻子收到了丈夫寄的信但是丈夫没有收到妻子回信的概率是多少？

解答：

第一步：妻子没有给丈夫回信有两种情况：1. 丈夫的信寄丢了（1/3的概率）2. 丈夫的信没寄丢（2/3的概率）但是妻子的回信寄丢了（1/3的概率）

第二步：题目问的是第2种情况发生的概率 - 丈夫的信没有寄丢是既定事实的这种情况 - 占全部可能性的比例是多少？

第三步：（2/3*1/3）/（1/3+2/3*1/3）= 2/5

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