Explanation of the answers to the three questions

Answers to three questions

A brief introduction to the three door problem

First , Introduce what is the three door problem . Here you are 3 A box , There is a box with a prize , The other two are empty boxes . We can't see any difference from the outside . Let's choose one at random first , Next , The host opens an empty box , Now do you want to change the box you originally chose ？

The answer is to change the box ！

The probability of winning the prize after changing the box is 2/3, The probability of not changing is 1/3.

Is it quite different from our intuition ？ The host opens a box , One of the two boxes has a prize , Then our winning rate should not be all 1/2 Do you ？

Tell the truth , The child thought so at the beginning , I didn't understand it even after others gave an answer （ Stupid to death ...） Know that the child wrote a simulation of three questions with the idea of refuting them python experiment ,, The answer really should be changed , Slap your face .

The right way of thinking

When we first selected a box , Our winning probability is 1/3, That's good , But the problem is that we instinctively think that after the host opens a box, the probability of originally belonging to the box will be equally divided into two unopened boxes , The winning rate of these two boxes is 1/2. But is it really OK ？ The answer is No .

When we choose a box , The state of this box is different from the other two , We can 3 The boxes are divided into two parts , What we have chosen and what we have not chosen . The winning rate of the selected part is 1/3, The median rate of the unselected part is 2/3, It's obvious . The point is that the host opened a box in the unselected part , Theoretically, it should not affect the part we choose . The winning rate of the selected part is still 1/3, The unselected part of the median rate is still 2/3, But the box that was opened was originally 1/3 The winning rate becomes 0, The winning rate of unopened boxes ranges from 1/3 Change into 2/3.（ It will be difficult to understand here , But it is also the key to this problem .）

So now the situation is , Two boxes , The winning rate we chose is 1/3, The winning rate of the selected one is 2/3. So of course I have to change the box .

Code

I know the expression of words is difficult to understand , After all, I didn't change my inherent paranoia at the beginning , So let's look at the code .

Here are the running results ：

Windows PowerShell
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PS C:\Users\hunzi> & C:/Users/hunzi/AppData/Local/Programs/Python/Python310/python.exe c:/Users/hunzi/Desktop/ Three door problem simulation program .py
 No change   Win the prize ：1652
 No change   I didn't win ：3376
 in   Win the prize ：3342
 in   I didn't win ：1630
 The winning rate of not changing is ：0.32856006364359586 And the winning rate is ：0.672164119066774
PS C:\Users\hunzi> 

link ：https://pan.baidu.com/s/13lsx6NN03SsXwfT-lEtzeA
Extraction code ：1234

Avoid possible errors in the copy and paste process , The child is directly put in the net tray , Permanent validity . This is also my first time to try Baidu online disk sharing , Always try something new .