## 2009年3月3日星期二

### 從 Monty Hall problem 看維基百科

Monty Hall problem 是一道以一位美國遊戲節目主持人 Monty Hall 為名，非常著名的概率謎題：

1) 該文章依然有錯，而且還是在貢獻者明說沒有錯的情況下犯錯。這道謎題有一個通俗的解答：
Often the next explanation is given: Players initially have a 1/3 chance of choosing the car and a 2/3 chance of choosing the goat. Players who stick to their original choice therefore have only a 1/3 chance of winning the car (and a 2/3 chance of getting a goat). Players who switch always get the opposite of their original choice so they have a 2/3 chance of getting a car (and 1/3 chance of getting a goat).

This reasoning applies to all players at the start of the game without regard to which door the host opens, specifically before the host opens a particular door and gives the player the option to switch doors ...

... Although the reasoning above is correct it doesn't answer the precise question posed by the problem, which is whether a player should switch after being shown a particular open door.
Well, 我應該怎樣評價維基百科這個評價呢？或者這樣說：
Although Wikipedia correctly points out that the commonplace reasoning is misplaced, it hasn't noticed that such reasoning is in fact wrong. Consider an alternative scenario in which the host opens Door 3 whenever possible. Then the same reasoning applies and it still gives the same probability of winning by switching as 2/3. However, as mentioned in the section "Sources of confusion", when the player chooses Door 1 and the host opens Door 3 in this case, the probability of winning by switching is only 1/2.

The commonplace reasoning is wrong because it doesn't take into account how the host behaves, so it cannot explain why the answer is 2/3 in one case but 1/2 in another.

2) 即使是數理科目，也牽涉文化歷史，而有關的討論就像其他文化或政治討論一樣，難以說服所有人。結果最常見的情形，就是不管當中有多少問題，大部份人認同的想法都會被當成正統。前述的「煲冬瓜-ism」是一個例子，而這裏有問題的是 Bayesian probability。文章用 Bayes Theorem 來計算謎題的答案時這樣說：
In Bayesian terms, a probability P(A|I) is a number in [0,1] associated to a proposition A. The number expresses a degree of belief in the truth of A, subject to whatever background information I happens to be known.

3) 由於人人都可以修改，維基百科文章除了很難保持通順之外，亦很難對內容分輕重。以 Monty Hall problem 的解答為例，最簡明的其實是用了決策樹 (decision tree) 的那個解答，而文章展示的那個圖解，其實相當難明。然而一般人都有以圖為先的傾向，因此明明是較差的解，卻放在較重要的位置。