数学

[人狼確率論]人狼積分現る!

ユキ
ユキ
人狼ゲームはご存知ですか?どうも,ユキです。

今回は,人狼ゲームで,村人が連続でつられる確率は,\(\int_{k=1}^{n}sin^nx dx\)という積分を使って表すことが出来る、ということを話していこうと思います。

これまでも人狼ゲームに関する記事を書いているので,よかったら読んでください。

[人狼確率検証] n人村で人狼が勝つ確率 どうも,ユキです。 今回はパーティーゲームで、お馴染みの人狼ゲームについての検証記事です。 \(n\)人村で,人狼が...

人狼ゲームで,村人が\(m\)日連続でつられる確率

せっかくですから,問題形式を作ってみました。

ですが,この問題はそこそこ難しいので,結果だけ知りたいという人は,まとめだけでも見ていってください。

自作問題

 

問題1:次の文は,人狼ゲームにまつわる問いである。問題文を読んで次の問いに答えよ

人狼が1人で,村人が\(n-1\)の合計\(n\)人の\(n\)人村を考える。人狼は毎日,1日の終わりに1人ずつ捕食するので,人狼を処刑したい。そこで,\(n\)人村の住民は毎日,昼に集まって,1人ずつランダムに処刑することにした。

ただし,処刑は,人狼がいなくなるまでの間だけ続き,「占い師」,「霊能者」,「狂人」,「ハンター」など特殊な役職についている人はいないものとする。

 

(1) 1日目の昼に,村人が処刑される確率\(a_1\)をもとめよ。

 

(2) 2日目の昼に,村人が処刑される確率\(a_2\)をもとめよ

 

(3) \(k\)日目の昼に,村人が処刑される確率\(a_k\)をもとめよ。

 

(4) \(n\)が奇数のとき,

$$\int_{0}^{\frac{\pi}{2}} sin^nx dx=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4} \cdots \frac{4}{5}\frac{2}{3}1$$

となり,\(n\)が偶数のとき,

$$\int_{0}^{\frac{\pi}{2}} sin^nx dx=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4} \cdots \frac{3}{4}\frac{1}{2}\frac{\pi}{2}$$

となることを部分積分を用いて示せ。

 

(5) (4)の結果を用いて,\(k\)日目の昼に,村人が処刑される確率\(a_k\)を積分の形で表せ。ただし,\(n\)は奇数とし,\(n>2k\)とする。

 

(6) \(n\)が偶数である場合の\(a_k\)を積分の形で表せ,ただし,\(n>2k+1\)とする。

 

 

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自作問題の解答

解答(1)

\(n\)人の中から,村人\(n-1\)人の誰かが処刑されればよいので,

$$a_1=\frac{n-1}{n}$$

解答(2) 

1日目の昼に村人が処刑され,かつ2人目に村人が処刑されればよい。

ただし,1日目と2日目で\(n\)人村から村人2人減っていることを考慮すると,2日目の\(n\)人村人の人数は,\(n-1-1=n-2\)人となる。

 

よって,1日目の昼に村人が処刑され,かつ2人目に村人が処刑される確率\(a_2\)は,

$$a_2=a_1\times \frac{n-3}{n-2}=\frac{n-1}{n} \frac{n-3}{n-2}$$

 

解答(3)

確率\(a_k\)は,1日目から\(k\)日目まで村人が処刑され続ける確率なので,

$$a_k=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4} \cdots \frac{n-2k+1}{n-2k+2}$$

 

 

解答(4) 

\(I_n=\int_{0}^{\frac{\pi}{2}} sin^n x dx\)とおくと,

$$ I_n=\int_{0}^{\frac{\pi}{2}}sin^{n-1} x sin xdx $$

ここで,部分積分を使うと,

$$=\int_{0}^{\frac{\pi}{2}}sin^{n-1} x (-cos x) dx $$

$$=[ sin^{n-1} x (-cos x) ]_{0}^\frac{\pi}{2} -\int_{0}^{\frac{\pi}{2}}( sin^{n-1} x )’ (-cos x) dx $$

$$=0-(n-1)\int_{0}^{\frac{\pi}{2}}sin^{n-2} x (cos x)(-cos x) dx $$

$$=(n-1)\int_{0}^{\frac{\pi}{2}}sin^{n-2} x (cos^2 x) dx $$

$$=(n-1)\int_{0}^{\frac{\pi}{2}}sin^{n-2} x (1- sin^2 x) dx $$

$$=(n-1)\int_{0}^{\frac{\pi}{2}}sin^{n-2} x dx- (n-1)\int_{0}^{\frac{\pi}{2}}sin^{n} x dx $$

ここで,\( I_n=\int_{0}^{\frac{\pi}{2}} sin^n x dx \)\( I_{n-2}=\int_{0}^{\frac{\pi}{2}} sin^{n-2} x dx \)より,

$$=(n-1)I_{n-2}-(n-1)I_n$$

元の式が\(I_n\)なので,

$$I_n=(n-1)I_{n-2}-(n-1)I_n $$

$$I_n=\frac{n-1}{n} I_{n-2}$$

\(n\)が奇数であることを利用すると,\(I_n\)は,

$$ I_n=\frac{n-1}{n} \frac{n-3}{n-2} I_{n-4}$$

$$ I_n=\frac{n-1}{n} \frac{n-3}{n-2} \frac{n-5}{n-4}I_{n-6}$$

$$ I_n=\frac{n-1}{n} \frac{n-3}{n-2} \frac{n-5}{n-4} \cdots \frac{4}{5} \frac{2}{3}I_1$$

ここで,\(I_1\)について,

$$I_1=\int_{0}^{\frac{\pi}{2}} sin x dx=1$$

より,

$$\int_{0}^{\frac{\pi}{2}} sin^nx dx=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4} \cdots \frac{4}{5}\frac{2}{3}1$$

 

\(n\)が偶数の場合は,先ほど同様に式変形をすると,

$$ I_n=\frac{n-1}{n} \frac{n-3}{n-2} \frac{n-5}{n-4} \cdots \frac{3}{4}\frac{1}{2}I_0$$

ここで,\(I_0\)について,

$$I_0=\int_{0}^{\frac{\pi}{2}} 1 dx=\frac{\pi}{2}$$

より,

$$ I_n=\frac{n-1}{n} \frac{n-3}{n-2} \frac{n-5}{n-4} \cdots \frac{3}{4}\frac{1}{2}\frac{\pi}{2}$$

<>

 

解答(5)

\(a_k\)を次のように変形させる

 

$$a_k=\frac{n-1}{n}\frac{n-3}{n-2}・\cdots・\frac{n-2k+1}{n-2k+2} \frac{n-2k-1}{n-2k}・\cdots・\frac{2}{3}1$$

$$÷\frac{n-2k-1}{n-2k}・\cdots・\frac{2}{3}1$$

 

そうすると,解答(4)で用いた\(I_n\)を使って\(a_k\)が表せる

$$a_k=I_n÷I_{n-2k}$$

よって,\(a_k\)は,

$$a_k=\int_{0}^{\frac{\pi}{2}} sin^nx dx÷\int_{0}^{\frac{\pi}{2}}sin^{n-2k}xdx$$

 

解答(6)

\(a_k\)を変形すると,

$$a_k=\frac{n-1}{n}\frac{n-3}{n-2}・\cdots・\frac{n-2k+1}{n-2k+2} \frac{n-2k-1}{n-2k}・\cdots・\frac{3}{4}\frac{1}{2}\frac{\pi}{2}$$

$$÷\frac{n-2k-1}{n-2k}・\cdots・\frac{3}{4}\frac{1}{2}\frac{\pi}{2}$$

 

\(I_n\)を使って,\(a_k\)が表せる

$$a_k=I_n÷I_{n-2k}$$

 

よって,\(a_k\)は,

$$a_k=\int_{0}^{\frac{\pi}{2}}sin^nx dx÷\int_{0}^{\frac{\pi}{2}} sin^{n-2k}x dx $$

 

\(n\)人村,1人狼のとき、人狼が勝つ確率を\(a_n\)とする。

(1)\(n\)が奇数のとき,

\(a_n=\frac{(n-1)!!}{n!!}\)

(2)\(n\)が偶数のとき

\(a_n=2\frac{(n-1)!!}{n!!}\)

 

と,前回の記事でもとめていた。これらを積分の形で表すと,

(1)\(n\)が奇数のとき,

\(a_n=\int_{0}^{\frac{\pi}{2}} sin^nx dx\)

(2)\(n\)が偶数のとき

\(a_n=\pi \times \int_{0}^{\frac{\pi}{2}} sin^nx dx\)

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