估中有獎

我在兩張卡片上分別寫下兩個不同的實數 (Real Number)，然後你選其中一張，我讓你看看你選的卡片上面的數字。現在，你估另一張卡片上的數字是較大還是較小？

試想出一個策略，使得無論我寫下那兩個實數，你估中的機會都大於 1/2。

(e.g. 估中的話在 MD Academic seminar 中有優先座位選擇權 ^^)

驟眼看，也許你會想：(車！) 我怎樣估也只得 1/2 機會估中。此策略沒理由存在！

但我告訴你，它是存在的！ 看答案前再想一想吧。

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其中一個策略如下：

Construct a strictly increasing function f(x) from the real line to the open interval (0, 1).
For example, f(x) can be
or (the latter is the cumulative distribution of a normal random variable), etc. Can you create another example for f(x)?

The strategy is simple:

If the number you picked is x, guess that it is larger with probability f(x).

Here' s how you can do it "practically": After you see the number, say x, create a coin which has probability f(x) to show up head and 1-f(x) to show up tail, and then flip it. Then you guess "x is bigger" if head shows up, and guess "x is smaller" if tail shows up.

Suppose the number I wrote are a and b, where a < b, then the probability that you guess correctly is

which is bigger than 1/2 as f(b)>f(a). ( Verify the formula above!)

The interesting thing in the above strategy is:
Even if you look at the number x, you do not know whether you'll guess "larger" or "smaller" until you've flipped the coin. This is a so called "Probabilistic Strategy".

若你想知多一點有關概率的問題，萬勿錯過 12 月 28 日（星期日）的 MD Academic seminar !


p.s. 此問題與 MD Academic seminar 無關，如有雷同，實屬巧合。