【数学A】例題2.2.8：さいころの目の最大値・最小値（One More）★★★

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% 例題A2.2.8：さいころの目の最大値・最小値（One More）★★★
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3個のさいころを投げるとき，次の確率を求めよ．

(1) 出る目の最大値が4以下である確率

(2) 出る目の最大値が4である確率

(3) 出る目の最大値が4，最小値が2である確率

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% 例題A2.2.8の解答（One More）★★★
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(1) 目の最大値が4以下であるためには，3個のさいころの目がすべて$1,2,3,4$のいずれかであればよい．

よって，求める確率は，$(\frac{4}{6})^3=(\frac{2}{3})^3=\frac{8}{27}\cdots(\mathrm{i})$

(2) 目の最大値が4となるのは，目の最大値が4以下である場合から，目の最大値が3以下である場合を除いた場合である．

目の最大値が3以下となる確率は，$(\frac{3}{6})^3=(\frac{1}{2})^3=\frac{1}{8}\cdots(\mathrm{ii})$

よって，(i)，(ii)より，求める確率は，$\frac{8}{27}-\frac{1}{8}=\frac{37}{216}$

(3) 3個のさいころの目が，

すべて$2,3,4$のいずれかである事象を$A$，

すべて$3,4$のいずれかである事象を$B$，

すべて$2,3$のいずれかである事象を$C$

とすると，求める確率は，

\begin{align*}
P(A)-P(B\cup C)&=P(A)-\{P(B)+P(C)-P(B\cap C)\}\\
&=(\frac{3}{6})^3-\{(\frac{2}{6})^3+(\frac{2}{6})^3-(\frac{1}{6})^3\}\\
&=\frac{1}{18}
\end{align*}

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% 問題A2.2.8：さいころの目の最大値・最小値（One More）★★★
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4個のさいころを投げるとき，次の確率を求めよ．

(1) 出る目の最小値が4以上である確率

(2) 出る目の最小値が4である確率

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% 問題A2.2.8の解答（One More）★★★
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(1) 目の最小値が4以上であるためには，4個のさいころの目がすべて$4,5,6$のいずれかであればよい．

よって，求める確率は，$(\frac{3}{6})^4=(\frac{1}{2})^4=\frac{1}{16}\cdots(\mathrm{i})$

(2) 目の最小値が4となるのは，目の最小値が4以上である場合から，目の最小値が5以上である場合を除いた場合である．

目の最小値が5以上である確率は，$(\frac{2}{6})^4=(\frac{1}{3})^4={\frac{1}{81}}\cdots(\mathrm{ii})$

よって，(i)，(ii)より，求める確率は，$\frac{1}{16}-\frac{1}{81}=\frac{65}{1296}$

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% 例題A2.2.8の側注：さいころの目の最大値・最小値（One More）★★★
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% 側注1
最大値が$k$となる確率は，最大値が$k$以下の確率から$k-1$以下の確率を引く．

% 側注2
\begin{align*}
&\text{(最大値が4の確率)}\\
&=\text{(最大値が4以下の確率)}\\
&-\text{(最大値が3以下の確率)}
\end{align*}

% 側注3
$P(B\cap C)$は3個のさいころがすべて3となる確率であるから，$(\frac{1}{6})^3$

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% 問題A2.2.8の側注：さいころの目の最大値・最小値（One More）★★★
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% 側注1
最小値が$k$となる確率は，最小値が$k$以上の確率から$k+1$以上の確率を引く．

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% 例題A2.2.8の考え方・補足など：さいころの目の最大値・最小値（One More）★★★
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% 考え方
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(1) 最大値が4以下の場合であるから，さいころの目がすべて$1,2,3,4$のいずれかの目が出る場合を考えればよい．

(2) (1)の場合は，さいころが「1，1，2」や「1，3，2」など，最大値が3以下の場合が含まれている（最大値が4ではない）．そこで，(1)から最大値が3以下である場合を除いたものを考える．

(3) 最大値が4，最小値が2であることから，すべて2，3，4の目が出る場合から，2と4の目が少なくとも1回ずつ出る確率を考える．つまり，2，3，4の目が出る場合から，2または4の目が出ない確率を除いたものを求める．

(2)

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最大値が$k$となる確率は，最大値が$k$以下の確率から$k-1$以下の確率を引く．