政局的盒子

春末夏初，政治局领导（简称政局）下基层看望西林人民，随身带来3个盒子，其中一个盒子里有一朵象征领导关怀的鲜花。领导让西林人民代表挑一个盒子, 代表很从容地挑了一个，心想，就是挑不中西林人民也不会怪罪，三分之一的机会嘛谁都知道。之后政局从西林代表没挑中的两个盒子中，拿走了一个空盒子，然后指着那个剩下的盒子，问西林代表：现在给你个机会，你可以把你原来挑中的那个盒子，换成这个盒子，你换不换？ 

得到领导的关怀对于西林人民至关重要，代表没了底，急切地把询问的目光投给了身后的西林人民。 

西林一下子就炸开了锅，换还是不换？经过几番争辩，在西林识字份子中形成了两派意见： 

67派（少数派）： 认为要换，因为换了会有更高的机会（67％概率）得到领导关怀的鲜花。 

50派（多数派）：认为换不换无所谓，代表你随便，挑不中我们不会怨你，而且，这派的很多人认为，你多余来问我们，浪费大家的时间，因为，现在是两个盒子一个有，一个没有，就算你说破大天，从两个盒子里选中一个有花的机会怎么都是50％。 

您是属于哪一派？






以下是两派的主要争论根据（部分）

67派

 Bill,

We have differences, but also have consensus.

I agree with you, the condition of the probability changed once you remove one box. Yet our difference is you believe the condition change applies equally to the 2 boxes left (A and B), but I believe that only applies to the box not picked by you (B and C).

No matter how many times Box A (the box you picked) is empty, it will never be put away. However, Box B, C (the boxes not picked by you) have the probability to be picked and put away if they are empty, but will remain in the game (be the eventual Box B) if not picked or put back if there is flower in it. 

So the condition change for the remaining boxes only applies to the box not picked by you. 

There will be no Box B until the following 2 steps are accomplished: 

Step 1. You pick a box (Box A)

Step 2. Zheng Ju put away a box that is empty (Box C). 

Step 2 is conditional, i.e.  in case Zheng Ju picked a box with the flower, he would put it back and pick the other box (for sure would be empty) instead.

That is why all 3 boxes had same probability initially - i.e. 1/3, but after the condition change (Zheng Ju only put away empty box), the box left in the game (Box B) has the higher probability to have the flower. 

If you are crazy (as we all are) and want to test, that box's (so called Box B) probability ranges between 1/3 to 2/3, depending on how many times you try. If zillion times, I bet probability to be 2/3.

It can also help us if we agree with the following:

What is the chance of box B or C be picked and put away? -- 50% each, initially, However:

If B has flower and picked - B will be put back and remains to be B,

If C has flower and picked - C will be put back and becomes B.

If B has no flower and picked - B will be put away and becomes C.

If C has no flower and picked - C will be put away and remains to be C.

It is by such conditional change (1 box be put away, empty box only) taking effect after your pick 1 box (you call Box A), the probability change only happens in the 2 boxes  (B and C) you did not pick initially.

It may be easier to look at the probability of being empty. 

With C is conditioned as 100% empty, Box B's probability of being empty dropped from 2/3 to 1/3. 

But Box A's probability of being empty remains to be 2/3.-- Because Box A never joins the game  (being put away if it is picked and happens to be empty).

Eva

50派 

Eva,

I feel the beauty of the Round Table is that we have opportunity to share the way(s) we think. Actually, I always feel happier being convinced than convincing, because I learn something new when I am convinced.

I maybe totally wrong (talking probability), but this is how I think about this Zheng Ju's box as a paradox.

1) The key difference between us is, when second chance given (condition changed), whether Box A shares the benefit from eliminating Box C, or only Box B has the privilege of that benefit.

In other words, whether the 1/3 of probability of Box C goes to Box B only，just because I didn't choose Box B or C at first place, or evenly distributes to Box A and Box B.

2) Think this scenario from different perspective:

Three participants in the game now. Person #1 picked Box A; #2, B; #3, C. Each of them has equal probability (1/3) of having the flower.

Now, the Judge announces to Person #3:    I am sorry, you are out. The Box C you picked is empty. Person #3 is gone sadly with tears.

Then, Judge turns to Persons #1 and #2:     Congratulations! We are almost there. Now I give both of you another chance. Any of you, or both of you, want to switch box now? And tell me WHY.

According the Zheng Ju's Box 2/3 theory, both of them should switch to get the better chance (>50% or 67% of probability) to get the flower, and if you repeat this process many times, both of them will have better chance against the other to win.

That is why this is a paradox.

Bill