November 12, 2005
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令人R爆頭的 Monty Hall Problem
美國有一份《大觀雜誌》（Parade）曾經開了一個專欄叫「瑪麗蓮答客問」（Ask Marilyn），由瑪麗蓮‧沙文特（Marilyn vos Savant）執筆，雜誌上說她是「吉尼斯世界紀錄名人堂」中智商最高的人，她在專欄中回答讀者投書的數學問題。一九九○年九月，馬里蘭州哥倫比亞地區的讀者克雷格‧惠特克（Craig F. Whitaker）投書問了這麼一個問題
-------------------------------------------------------------------------------------------你參加一項電視遊戲節目，這個節目提供的獎品是一部汽車。節目主持人先給你看三扇門，說其中一扇門裡面是一部汽車，另外兩扇門裡面是山羊。他要你挑選一 扇門。你選了，呢個時侯仲門沒打開。主持人先打開你未挑選的兩扇門中的一扇，裡面是一頭山羊（因為他知道門後面是什麼），然後他說在那扇門打開之前你還有 最後一次機會可以改變主意，你可以得到一部汽車，否則就是一頭山羊。這時他問你為了以較大的概率選中汽車，要不要改變主意換另外一扇沒有打開的門。請問你 該怎麼辦？
Comments (12)
直覺沒有區別
無論第一次選的門後面是車是羊﹐其中有羊的門一定被會打開﹐所以是50%
打開後還是50%
haha it took me awhile to understand this one, but if you think about it as choosing from 50 doors and there are 49 goats, and the host show you 48 goats after your pick, then it makes sense.....
i think that was explaint in mark haddon's "the curious incident of the dog in the night time". it is exactly the same story that the author talked about i remember i read it. but i dun remember the theory. haha. i think it's better to change choice. not sure.
dollymushroom, are you saying that you should stick with your original choice?
i do not quite understand your logic. Say, there are 50 doors and the host show you 48 goats and ask you to switch or no. intuitively, it seems that you hit the right door. but then, the host is going to open 48 doors with goats anyway, i.e. his act is not conditional on your pick. think it this way, even if you pick the wrong door, the host will show you 48 goats, leaving you to ponder whether to switch or not.
exactly. in other words, you could have made the right choice all along. why change then?
even if you really made the wrong choice, there is no way to prove that until one of the remaining doors is opened. so why bother changing your choice?
essentially the whole thing boils down to just a head-or-tail game. So it doesn't matter whether you change your choice or not. But changing your choice means more wasted time and more emotions tied to the game. Not a very smart move.
actually no, i think you should always switch. My thinking was that originally you have 50 doors and there is only one sedan, assuming you randomly picked one, the chance that you have picked a goat is much higher than that of the sedan. so after the host show you the 48 goats, i thought it becomes logical to switch....i just thought it's easier to think about a larger number of doors so the contrast of the probability between picking a goat or a sedan is bigger...
改變主意應該會令選中汽車的機會提高
有答案嗎？
dollymushroom, oic. now i understand what you mean. although the host's act is not conditional on your pick, his act would eliminate much uncertainty. think it this way, there are 1 million doors. the probability of picking the right one is virtually zero, i.e. your pick is wrong for SURE. then the host eliminate 999998 doors for you. now there are two doors left, one with a goat and one with a sedan. then you ask yourself, "am i so lucky that i picked the right one from the outset?" the answer is clearly no.
so you should switch.
can i put it this way:
there are two sets of probability
1. (a). Prob(your pick is right)
(b). Prob(your pick is right | doors opened by the host)
2. (a) Prob(the remaining door is right)
(b). Prob(the remaining door is right | doors opened by the host)
1a = 1b (this is exactly what I was thinking)
but 2a is not equal to 2b. In fact, 2b >> 2a. so you should switch.
來看小弟xanga的都是高水平xan民呢。其實小弟認為也許「正統」的solution在數學上是對的(詳細解釋見英文版Wikipedia), 但實際上當你真的參加這樣的一個比賽的話, 無論你是PhD還是牛頭角師奶, 你也只會認為是五五波吧, 站在攝影機前, 被曾志偉嚇一嚇, 管它什麼conditional不conditional。
我有個問題 : 如果轉變會能機會率上升至2/3, 如這時加入一個未選過既參賽者, 佢選中既機會只係得 1/2. 咁係咪有些少奇怪咁呢?
不轉選擇, 贏的機會是 1/3.
轉選擇, 贏的機會是 2/3.
因為無論主持人早就知道答案, 故你揀o岩或揀錯佢都會開到一隻羊俾你睇. 換言之, 你轉的話, 你是估了三個選擇中的兩個, 不轉的話, 只是三個中的一個.
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