Quod Erat Demonstrandum

2008/09/01

利用對稱性解概率問題

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 11:17 下午

想收手,無奈自己按奈不住,趁開學還未忙死,偷偷又講數:

對稱性有時可以幫助解題,舉兩例。

例一(頗常見的技巧)
兩顆公平骰子,一顆是紅色,一顆是藍色。現獨立地各擲兩次,求紅色骰子的總點數大於藍色骰子的總點數之概率。

若設
X = 紅色骰子的總點數
Y = 藍色骰子的總點數
則,題目要求 P(X > Y)

直接列舉「X > Y」的情況比較煩,我們嘗試這樣想。

所有可能性不離以下三個:

「X = Y」,「X > Y」及「X < Y」。

易知「X > Y」及「X < Y」是所謂對稱的情況,只要想像一下,把紅藍兩色對調,便可以得到一一對應的關係。從而 P(X > Y) = P(X < Y)

於是,

P(X > Y) = \frac{1 - P(X = Y)}{2}

好了,現在計 P(X = Y)

X 可取值由 2 至 12,又用所謂對稱性,易知

P(X = 2) = P(X = 12) = \frac{1}{36}
P(X = 3) = P(X = 11) = \frac{2}{36}
P(X = 4) = P(X = 10) = \frac{3}{36}
P(X = 5) = P(X = 9) = \frac{4}{36}
P(X = 6) = P(X = 8) = \frac{5}{36}

還有

P(X = 7) = \frac{6}{36}

嗯,考慮比如

P(X = Y = 2) = P(X = 2)P(Y = 2) = (\frac{1}{36})^2

易得

P(X = Y) = 2\{(\frac{1}{36})^2 + (\frac{2}{36})^2 + \dots + (\frac{5}{36})^2\} + (\frac{6}{36})^2

從而不難得出 P(X > Y)。(我懶,不計了。)

例二(比較少見)

在區間 (a , b) 中隨意獨立地取 n 點,求相距最遠的兩點之間的距離之期望值(expectation)。

解題的關鍵又是對稱性:每個點有著相等的分佈(distribution)情況。

n 個點把 (a , b) 分成 n + 1 部份,設各部份之長度依次為 X_1, X_2, \dots, X_{n + 1}

X_1, X_2, \dots, X_{n + 1}n + 1 個隨機變量有同等的分佈,故

E(X_1) = E(X_2) = \dots = E(X_{n + 1}) – – – – – – (*)

另外,易知

X_1 + X_2 + \dots + X_{n + 1} = b - a

E(X_1) + E(X_2) + \dots + E(X_{n + 1}) = b - a

由 (*),得 E(X_i) = \frac{b - a}{n + 1} (i = 1, 2, \dots, n + 1)

現在,相距最遠的兩點之間的距離 = X_2 + X_3 + \dots + X_n

所以,相距最遠的兩點之間的距離之期望值 = (n - 1)E(X_i) = \frac{(b - a)(n - 1)}{n + 1}

習題

一副 52 張的普通樸克,當中有 4 張 A。隨意洗牌後,從最頂的一張牌開始,一張接一張翻牌,直至翻到第 3 張 A 為止。求被翻過的樸克牌數目之期望值。

[Hint:仿傚例二]

當然還有無數其他例子運用到所謂對稱性,歡迎提供,謝謝。

4 則迴響 »

  1. 我想問…

    你寫的數學式是用什麼軟件編出來的-.-?

    例子一很爽, 我也想不出來..也許我笨

    迴響 由 Humdrum — 2008/09/08 @ 8:34 下午 | 回覆

  2. Humdrum,

    you may read the following post

    http://faq.wordpress.com/2007/02/18/can-i-put-math-or-equations-in-my-posts/

    for details

    迴響 由 johnmayhk — 2008/09/09 @ 7:56 上午 | 回覆

  3. I find it much harder to “survive" in form 3…
    Form two I can get into the top ten doing my best, but I don’t think so now, I’ve gotta do more than I can, else eliminated.

    I never noticed there’s a RSS feed for comments on posts in your blog…

    迴響 由 Edmund — 2008/09/09 @ 8:44 下午 | 回覆

  4. Edmund, “doing your best" should be the best way to enjoy your schooling. Top ten or not, well, take it easy, at least, you are in the elite class.

    Though I’d written this blog for 1 year, I really don’t know what is RSS (so poor) and some functions of this blog, like “Blogroll", may be someone may tell me what are they. XP

    迴響 由 johnmayhk — 2008/09/10 @ 9:00 上午 | 回覆


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