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为什么不是2/9

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

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

老师我有两个问题：第一个问题是，题目明明问的是，妻子收到丈夫的信，但丈夫没收到妻子的信，妻子只有收到丈夫的信才会回信，所以第二个树杈是不存在的。为什么不是2/3*1/3=2/9？见如下图：

第二个问题是左红框为什么等于右红框？课程里的贝叶斯公式只有分子的推导，没有分母的推导。

嗨，爱思考的PZer你好：

第一个问题是，题目明明问的是，妻子收到丈夫的信，但丈夫没收到妻子的信，妻子只有收到丈夫的信才会回信，所以第二个树杈是不存在的。为什么不是2/3*1/3=2/9？

因为题目问的是在给定男人没有收到老婆来信的前提下，妻子收到信件的概率，您这里直接相乘并没有考虑给定的条件。因此不对。

第二个问题是左红框为什么等于右红框？课程里的贝叶斯公式只有分子的推导，没有分母的推导。

这就是个全概率公式，请参考这个

这个题如果同学比较纠结的话，我们另辟蹊径，换一种方法来做。

我们就假设男人给女人发了90次信件，那么根据题目所述，有60封女人是收到了，30封石沉大海。

然后女人收到这60封信，也会回复60次，那么其中40封男人收到，20封石沉大海。

也就说男人一共发了90次信，只收到40次回信，而50封信是收不到的，对于这50封收不到的信来说，分成2部分，其中30封是女人没收到，20封是女人收到信后的回复没发过去。因此是20/50=2/5选D

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为什么不是2/9

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