Quod Erat Demonstrandum

2009/02/03

出現 A 先於 B 的機會

常見題目:擲一顆公平骰子若干次,求得到 1 點先於得到 6 點的概率。

此例容易,因 1 點和 6 點在地位上無異,易知要求概率為 \frac{1}{2}

改一改題:擲兩顆公平骰子若干次,求得到總點數為 4 點先於得到總點數為 9 點的概率。

對某次投擲,易得 P(4) = \frac{3}{36}P(9) = \frac{4}{36},但同學往往難明白,為何題目要求的概率是

\frac{3}{36} \div (\frac{3}{36} + \frac{4}{36}) = \frac{3}{7}

其實,透過全概率公式便可。

在樣本空間 S 中,存在事件 A 及 B。設事件 C 為「出現 A 先於 B」。由全概率公式,得

P(C) = P(A)P(C | A) + P(B)P(C | B) + P(D)P(C | D)

其中 D = S \setminus (A \cup B)

化簡上式,

P(C) = P(A)\times 1 + P(B)\times 0 + P(D)P(C)
P(C) = \frac{P(A)}{1 - P(D)} = \frac{P(A)}{P(A \cup B)}

故此,

P(C) = \frac{P(A)}{P(A) + P(B) - P(A \cap B)} – – – – – – (*)

若 A 和 B 為互斥事件,P(A \cap B) = 0,(*) 進一步簡化為

P(C) = \frac{P(A)}{P(A) + P(B)}

利用上式,立即計算上述例子中,「出現 4 點先於 9 點」的概率為 \frac{3}{7}

3 則迴響 »

  1. I have a question, how can you make the pages such that it only shows the summary or the first few sentences of each post? (i.e. you have a little (more) button for me to click into this post)

    迴響 由 koopakoo — 2009/02/05 @ 12:31 下午 | 回覆

  2. Above the edit window, you may see a tool bar. Click the button “more" on the bar, then the tag < !–more– > will appear in you content of editing , and the content after the tag < !–more– > will be hidden just showing (more) in your post. But the effect will not be seen during editing, it works only after publishing your post. Try!

    迴響 由 johnmayhk — 2009/02/05 @ 2:01 下午 | 回覆

  3. It worked! Thank you very much.

    迴響 由 koopakoo — 2009/02/05 @ 8:14 下午 | 回覆


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