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妻子回信的概率

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

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

妻子回信的概率应该是100%

信件能否送达的概率是2/3

这两个不应该相等

嗨，爱思考的PZer你好：

题目说的是，在丈夫没有收到回信的前提下，问妻子收到信件的概率是多少？这是典型的条件概率问法，所以考察的是贝叶斯公式。

利用条件概率二叉树，丈夫没有收到回信的情况就两种：

丈夫的信没送给妻子，概率1/3；
丈夫的信送给妻子了，但丈夫没有收到妻子的回信，概率=2/3 * 1/3 = 2/9.

所以“丈夫没有收到回信” （事件B）的非条件概率P（B） = 1/3 + 2/9 = 5/9.

而"丈夫的信送到妻子手里（事件A）并且丈夫没有收到回信（事件B）" 这个非条件概率P(AB) = 2/3 * 1/3 = 2/9. 非条件概率就是直接把两个独立事件各自的概率相乘。

所以最后利用贝叶斯条件概率公式，在丈夫没有收到回信的条件下（已知B为条件），妻子收到信件（事件A发生）的概率P(A|B) = P(AB) / P(B) = 2/5.

题目说了妻子一定会回信，所以不需要考虑妻子回信的概率。

我们只需要考虑丈夫的信是否送到妻子，以及妻子的回信是否送到丈夫，这两个独立事件。信件送到对方手上的概率都是2/3，送不到的概率都是1/3。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

妻子回信的概率

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