富翁的遺產 - 拼圖

※ 引述《weselyong (Wesely翁)》之銘言：
: 今天早上聽同學講的
: 我不知道有沒有OP耶...
: 請再告訴我～
: ****************分隔線*******************
: 有一個非常富有的老翁臨死前把他的全部財產分成兩張支票
: 要給他兩個兒子，其中一張的面額是另一張的兩倍。
: 但是支票分別放在信封裡面。
: 老富翁說：
: 「你們看過之後可以決定要不要交換，前提是對方還不知道你這張的數字是多少的時候」
:
: 老大看到他拿到一千萬之後OS:
: 他那包比我大的機率是0.5，可能有兩千萬
: 比我小的機率是0.5，可能有五百萬
: 那我跟他換之後的期望值應該是1250萬！ 比我的1000萬多！當然換！
: 弟弟看到他拿到500萬之後也 OS一發：
: 哥哥可能拿到1000萬，機率是0.5
: 也可能只有250萬，機率是0.5
: 那我跟他換的期望值是 500 + 125 ＝ 625萬！比我的500萬多！當然換！
: 很明顯的其中必有詐
: 他們錯了嗎？
: （我不知道答案喔XDD 或者說我不確定我答案是對的）


這題維基百科裡有...


http://en.wikipedia.org/wiki/Two_envelopes_problem



1. I denote by A the amount in my selected envelope.
2. The probability that A is the smaller amount is 1/2, and that it is the
larger amount is also 1/2.
3. The other envelope may contain either 2A or A/2.
4. If A is the smaller amount the other envelope contains 2A.
5. If A is the larger amount the other envelope contains A/2.
6. Thus the other envelope contains 2A with probability 1/2 and A/2 with
probability 1/2.
7. So the expected value of the money in the other envelope is

{1 \over 2} 2A + {1 \over 2} {A \over 2} = {5 \over 4}A

8. This is greater than A, so I gain on average by switching.
9. After the switch, I can denote that content by B and reason in exactly
the same manner as above.
10. I will conclude that the most rational thing to do is to swap back
again.
11. To be rational, I will thus end up swapping envelopes indefinitely.
12. As it seems more rational to open just any envelope than to swap
indefinitely, we have a contradiction.

關鍵大致是兩個式子的"A"是不同情形下無法運算, 或是


Devlin writes:

To summarize: the paradox arises because you use the prior probabilities
to calculate the expected gain rather than the posterior probabilities. As we
have seen, it is not possible to choose a prior distribution which results in
a posterior distribution for which the original argument holds; there simply
are no circumstances in which it would be valid to always use probabilities
of 0.5.

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