【数学A】例題2.2.15：期待値（さいころの目）（One More）★

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% 例題A2.2.15：期待値（さいころの目）（One More）★
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2個のさいころを同時に投げるとき，出る目の差の絶対値の期待値を求めよ．

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% 例題A2.2.15の解答（One More）★
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2個のさいころをA，Bとすると，目の差の絶対値は，右の表のようになり，目の出方は$6\times 6=36(\text{通り})$

したがって，A，Bの出た目の差の絶対値を$X$とすると，$X$のとりうる値とそれぞれの値をとる確率は，下の表のようになる．

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline A$\backslash$B&1&2&3&4&5&6\\
\hline 1&0&1&2&3&4&5\\
\hline 2&1&0&1&2&3&4\\
\hline 3&2&1&0&1&2&3\\
\hline 4&3&2&1&0&1&2\\
\hline 5&4&3&2&1&0&1\\
\hline 6&5&4&3&2&1&0\\
\hline
\end{tabular}

\begin{tabular}{|c||c|c|c|c|c|c|c|}
\hline$X$&0&1&2&3&4&5&計\\
\hline 確率&$\frac{6}{36}$&$\frac{10}{36}$&$\frac{8}{36}$&$\frac{6}{36}$&$\frac{4}{36}$&$\frac{2}{36}$&1\\
\hline
\end{tabular}

よって，求める期待値は，

\[
0\times\frac{6}{36}+1\times\frac{10}{36}+2\times\frac{8}{36}+3\times\frac{6}{36}+4\times\frac{4}{36}+5\times\frac{2}{36}=\frac{35}{18}
\]

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% 問題A2.2.15：期待値（さいころの目）（One More）★
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2個のさいころを同時に投げるとき，出る目の和の期待値を求めよ．

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% 問題A2.2.15の解答（One More）★
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2個のさいころA，Bの出た目の和を$X$とすると，各目の出方に対する$X$の値は，右の表のようになり，目の出方は，$6\times 6=36\text{(通り)}$である．

右の表から，$X$のとりうる値とそれぞれの値をとる確率は，下の表のようになる．

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline A$\backslash$B&1&2&3&4&5&6\\
\hline 1&2&3&4&5&6&7\\
\hline 2&3&4&5&6&7&8\\
\hline 3&4&5&6&7&8&9\\
\hline 4&5&6&7&8&9&10\\
\hline 5&6&7&8&9&10&11\\
\hline 6&7&8&9&10&11&12\\
\hline
\end{tabular}

\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|}
\hline$X$&2&3&4&5&6&7&8&9&10&11&12&計\\
\hline 確率&$\frac{1}{36}$&$\frac{2}{36}$&$\frac{3}{36}$&$\frac{4}{36}$&$\frac{5}{36}$&$\frac{6}{36}$&$\frac{5}{36}$&$\frac{4}{36}$&$\frac{3}{36}$&$\frac{2}{36}$&$\frac{1}{36}$&1\\
\hline
\end{tabular}

よって，求める期待値は，

\begin{align*}
&2\times\frac{1}{36}+3\times\frac{2}{36}+4\times\frac{3}{36}+5\times\frac{4}{36}+6\times\frac{5}{36}+7\times\frac{6}{36}\\
&+8\times\frac{5}{36}+9\times\frac{4}{36}+10\times\frac{3}{36}+11\times\frac{2}{36}+12\times\frac{1}{36}\\
=&7
\end{align*}

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% 例題A2.2.15の側注：期待値（さいころの目）（One More）★
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% 側注1
表にまとめると場合の数を求めやすい．

% 側注2
確率の表は，約分しないでおくと期待値の計算が楽になる．また，確率の総和（計）が1になることを確認する．

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% 問題A2.2.15の側注：期待値（さいころの目）（One More）★
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% 側注1
表にまとめることで場合の数を数えやすくする．

% 側注2
確率の総和（計）が1になることを確認する．

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% 例題A2.2.15の考え方・補足など：期待値（さいころの目）（One More）★
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% 考え方
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$X$のとる値とその値になる確率が右の表のようなとき，$X$の期待値は，

\[
x_1p_1+x_2p_2+\cdots+x_np_n
\]

となることを利用する．

\begin{tabular}{|c||c|c|c|c|c|c|}
\hline$X$&$x_1$&$x_2$&$x_3$&$\cdots$&$x_n$&計\\
\hline 確率&$p_1$&$p_2$&$p_3$&$\cdots$&$p_n$&1\\
\hline
\end{tabular}

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% One Point
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期待値は，とり得る値にその確率を掛け，すべて加えて求める．