【数学A】例題2.1.1：確率の計算（One More）★

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次の確率を求めよ．

(1) 2個のさいころを投げるとき，目の和が6となる確率を求めよ．

(2) 3枚の硬貨を投げて，表2枚，裏1枚が出る確率を求めよ．

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(1) 目の出方は，
\[
6\times 6=36(\text{通り})
\]

目の和が6となるのは，

\[
(1,5),(2,4),(3,3),(4,2),(5,1)
\]
より，5通り

よって，求める確率は，$\frac{5}{36}$

(2) 3枚の硬貨を投げた場合に起こりうるすべての組み合わせは，

\[
2^3=8(\text{通り})
\]

このうち，表2枚，裏1枚が出る組み合わせは，$(\text{表，表，裏})$，$(\text{表，裏，表})$，$(\text{裏，表，表})$より，3通り

よって，求める確率は，$\frac{3}{8}$

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
和&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}

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% 問題A2.1.1：確率の計算（One More）★
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次の確率を求めよ．

(1) 2個のさいころを投げるとき，目の和が7となる確率を求めよ．

(2) 4枚の硬貨を投げて，表3枚，裏1枚が出る確率を求めよ．

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(1) 目の出方は，$6\times 6=36(\text{通り})$

目の和が7となるのは，$(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$より，6通り

よって，求める確率は，$\frac{6}{36}=\frac{1}{6}$

(2) 4枚の硬貨を投げた場合に起こりうるすべての組み合わせは，$2^4=16(\text{通り})$

このうち，表3枚，裏1枚が出る組み合わせは，$(\text{表，表，表，裏})$，$(\text{表，表，裏，表})$，$(\text{表，裏，表，表})$，$(\text{裏，表，表，表})$より，4通り

よって，求める確率は，$\frac{4}{16}=\frac{1}{4}$

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
和&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}

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% 例題A2.1.1の側注：確率の計算（One More）★
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% 側注1
表を作るとよい．例えば，$(1,2)$と$(2,1)$は別の出方として考える．

% 側注2
3枚の硬貨をA，B，Cと区別すると考えると，次のように表せる．
\[
(\underbrace{\text{表か裏}}_{\mathrm{A}},\underbrace{\text{表か裏}}_{\mathrm{B}},\underbrace{\text{表か裏}}_{\mathrm{C}})
\]

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% 側注1
表を作るとよい．

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起こりうるすべての場合の全体の集合$U$を定め，求めたい確率の事象$A$の根元事象を考える．このとき，求める事象$A$の起こる確率は，$P(A)=\frac{\text{(事象}A\text{の起こる場合の数)}}{\text{(起こりうるすべての場合の数)}}$となる．

なお，確率では，同じ形の硬貨，さいころ，くじ，カード，玉などの1つ1つを区別して考える（1つ1つを異なるものと考える）ことに注意すること．

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確率では，同じさいころ，硬貨でも区別して考える．

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(2)において，3枚の硬貨の表と裏の出方には，次の4つの場合があると考えられる．

(i) 表3枚

(ii) 表2枚，裏1枚

(iii) 表1枚，裏2枚

(iv) 裏3枚これらより，求める確率を4つの場合のうちの(ii)の1つの場合であるから，$\frac{1}{4}$としてしまうのは誤りであるので注意すること．例えば，(ii)と(iv)は同じ程度に起こることは期待できず，同様に確からしいということができない（数が大きい例を考えるとわかりやすい．300枚の硬貨を投げるとき，200枚表，100枚裏が出ることと，300枚裏が出ることは同じ程度に期待はできない）．

ある試行において根元事象のどれが起こることも同じ程度に期待できるとき，これらの根元事象は同様に確からしいという．3枚の硬貨を区別すると考え，表と裏の出方を$2^3=8(\text{通り})$とすることにより，同様に確からしいと考えることができる．