Re: [推理] 富翁的遺產

看板puzzle (益智遊戲 - 數獨,拼圖,推理,西洋棋)作者meowth (喵貓)時間14年前 (2010/10/23 15:52)推噓8(8推 0噓 4→)

留言12則, 7人參與討論串2/19 (看更多)

※ 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. -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 61.228.24.179