【数学A】例題2.2.5：3つの事象に関する反復試行の確率（One More）★★★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A2.2.5：3つの事象に関する反復試行の確率（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

赤玉1個，白玉2個，青玉3個が入っている袋の中から，1個の玉を取り出し，色を調べてからもとに戻すことを5回行うとき，次の確率を求めよ．

(1) 赤玉が1回，白玉が2回，青玉が2回出る確率

(2) 赤玉と白玉が出る回数が同じである確率

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A2.2.5の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

この袋から玉を1個取り出すとき，赤玉，白玉，青玉が出る確率は，それぞれ$\frac{1}{6},\frac{1}{3},\frac{1}{2}$である．

(1) 求める確率は，

\[
\frac{5!}{1!2!2!}\times(\frac{1}{6})^1(\frac{1}{3})^2(\frac{1}{2})^2=\frac{5}{36}
\]

(2) 赤玉と白玉が出る回数が同じであるのは，赤玉と白玉の出る回数が0回，1回，2回の3つの場合がある．

(i) 赤玉と白玉が1回も出ないとき

\[
(\frac{1}{2})^5=\frac{1}{32}
\]

(ii) 赤玉と白玉が1回ずつ出るとき

\[
\frac{5!}{1!1!3!}\times(\frac{1}{6})^1(\frac{1}{3})^1(\frac{1}{2})^3=\frac{5}{36}
\]

(iii) 赤玉と白玉が2回ずつ出るとき

\[
\frac{5!}{2!2!1!}\times(\frac{1}{6})^2(\frac{1}{3})^2(\frac{1}{2})^1=\frac{5}{108}
\]

よって，(i)〜(iii)より，求める確率は，

\[
\frac{1}{32}+\frac{5}{36}+\frac{5}{108}=\frac{187}{864}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題A2.2.5：3つの事象に関する反復試行の確率（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

赤玉1個，白玉2個，青玉2個が入っている袋の中から，1個の玉を取り出し，色を調べてからもとに戻すことを5回行うとき，次の確率を求めよ．

(1) 赤玉が1回，白玉が2回，青玉が2回出る確率

(2) 赤玉が出る回数が白玉が出る回数よりも1回だけ多くなる確率

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題A2.2.5の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

この袋から玉を1個取り出すとき，赤玉，白玉，青玉が出る確率は，それぞれ$\frac{1}{5},\frac{2}{5},\frac{2}{5}$である．

(1) 求める確率は，

\[
\frac{5!}{1!2!2!}\times(\frac{1}{5})^1(\frac{2}{5})^2(\frac{2}{5})^2=\frac{96}{625}
\]

(2) 赤玉が出る回数が白玉が出る回数よりも1回だけ多くなるのは，以下の3つの場合がある．

(i) 赤玉が1回，白玉が0回，青玉が4回出るとき

\[
\frac{5!}{1!4!}\times(\frac{1}{5})^1(\frac{2}{5})^4=\frac{16}{625}
\]

(ii) 赤玉が2回，白玉が1回，青玉が2回出るとき

\[
\frac{5!}{2!1!2!}\times(\frac{1}{5})^2(\frac{2}{5})^1(\frac{2}{5})^2=\frac{48}{625}
\]

(iii) 赤玉が3回，白玉が2回，青玉が0回出るとき

\[
\frac{5!}{3!2!}\times(\frac{1}{5})^3(\frac{2}{5})^2=\frac{8}{625}
\]

よって，(i)〜(iii)より，求める確率は，

\[
\frac{16}{625}+\frac{48}{625}+\frac{8}{625}=\frac{72}{625}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A2.2.5の側注：3つの事象に関する反復試行の確率（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
5回とも青玉が出る（赤玉0回，白玉0回）．

% 側注2
${}_5\mathrm{C}_1\times{}_4\mathrm{C}_1\times{}_3\mathrm{C}_3$でもよい．

% 側注3
5回のうち赤玉が2回，白玉が2回，青玉が1回出る場合の数は，$\frac{5!}{2!2!1!}$通りである（同じものを含む順列）．

% 側注4
5回のうち赤玉が1回，白玉が2回，青玉が2回出る場合の数は，$\frac{5!}{1!2!2!}$通りである（同じものを含む順列）．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題A2.2.5の側注：3つの事象に関する反復試行の確率（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
5回のうち赤玉が1回，白玉が0回，青玉が4回出る場合の数は，$\frac{5!}{1!0!4!}$通りである．

% 側注2
5回のうち赤玉が1回，白玉が2回，青玉が2回出る場合の数は，$\frac{5!}{1!2!2!}$通りである．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A2.2.5の考え方・補足など：3つの事象に関する反復試行の確率（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 考え方
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3つの事象に関する反復試行を考えればよい．(1)は，赤玉，白玉，白玉，青玉，青玉の同じものを含む順列であり$(\frac{5!}{1!2!2!}\text{通り})$，反復試行を考える．(2)は，(i)赤玉と白玉が1回も出ないとき，(ii)赤玉と白玉が1回ずつ出るとき，(iii)赤玉と白玉が2回ずつ出るときの3つの場合に分けて考える．