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% 基本事項A2.1.1：事象と確率（One More）
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(1) さいころを投げる，トランプのカードを引く，ルーレットを回すなど同じ条件のもとで繰り返し行うことのできる実験や観察のことを試行という．
(2) 試行の結果生じた事柄，現象のことを事象という．
(3) 試行によって生じる1つ1つの結果に対応する，これ以上分けることのできない事象を根元事象という．
(4) すべての根元事象を合わせた事象，すなわち，起こりうるすべての場合からなる事象を全事象という．
(5) 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．
(6) ある試行において，根元事象はすべて同様に確からしいとする．全事象$U$に含まれる根元事象の個数を$n(U)$，事象$A$に含まれる根元事象の個数を$n(A)$とするとき，事象$A$の起こる確率$P(A)$は，
\[
P(A)=\frac{n(A)}{n(U)}=\frac{\text{(事象A\text{の起こる場合の数})}}{\text{(起こりうるすべての場合の数)}}
\]
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% 基本事項A2.1.2：確率の基本性質（One More）
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(1) 事象$A$の確率の範囲$\cdots 0\leqq P(A)\leqq 1$
(2) 全事象$U$の確率$\cdots P(U)=1$，空事象$\varnothing$の確率$\cdots P(\varnothing)=0$
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% 基本事項A2.1.3：積事象と和事象，排反事象（One More）
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(1) 全事象$U$の部分集合$A,B$について，$A$と$B$の積事象$A\cap B$は「$A$と$B$がともに起こる」という事象であり，和事象$A\cup B$は「$A$または$B$が起こる」という事象である．
(2) 2つの事象$A,B$が同時には決して起こらない，すなわち，$A\cap B=\varnothing$のとき，事象$A,B$は互いに排反である，または，互いに排反事象であるという．
(3) 和事象の確率
\[
P(A\cup B)=P(A)+P(B)-P(A\cap B)
\]
また，事象$A,B$が互いに排反であるとき，次の確率の加法定理が成り立つ．
\[
P(A\cup B)=P(A)+P(B)
\]
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% 基本事項A2.1.4：余事象の確率（One More）
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事象$A$に対して，$A$が起こらない事象のことを余事象といい，$\overline{A}$で表す．
\[
P(\overline{A})=\frac{(\text{起こりうるすべての場合の数})-(\text{事象}A\text{の起こる場合の数})}{(\text{起こりうるすべての場合の数})}=1-P(A)
\]
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% 基本事項A2.2.1：独立な試行とその確率（One More）
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(1) 2つの試行$\mathrm{T}_1,\mathrm{T}_2$について，試行の結果が互いに影響し合わないとき，試行$\mathrm{T}_1,\mathrm{T}_2$は独立であるという．
(2) 2つの独立な試行$\mathrm{T}_1,\mathrm{T}_2$について，$\mathrm{T}_1$で事象$A$が起こり，$\mathrm{T}_2$で事象$B$が起こるという事象を$C$とすると，
\[
P(C)=P(A)\times P(B)
\]
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% 基本事項A2.2.2：反復試行の確率（One More）
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同じ条件のもとで同じ試行を繰り返し，それらの試行が独立であるとき，これを反復試行という．
1回の試行で事象$A$の起こる確率を$p$とすると，この試行を$n$回繰り返し行うとき，事象$A$がちょうど$r$回起こる確率は，
\[
{}_n\mathrm{C}_rp^r(1-p)^{n-r}
\]
ただし，$p^0=1,(1-p)^0=1$とする．
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% 基本事項A2.2.3：条件付き確率（One More）
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ある試行における2つの事象$A,B$について，事象$A$が起こったという条件のもとで事象$B$が起こる確率を，$A$が起こったときの$B$の条件付き確率といい，$P_A(B)$と表す．このとき，
\[
P(A\cap B)=P(A)P_A(B)(P_A(B)=\frac{P(A\cap B)}{P(A)})
\]
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% 基本事項A2.2.4：期待値（One More）
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ある試行の結果に応じて，$X$の値が$x_1,x_2,\ldots,x_n$のいずれかに定まり，それぞれの値となる確率が$p_1,p_2,\ldots,p_n$であるとする．このとき，
\[
E(X)=x_1p_1+x_2p_2+\cdots+x_np_n
\]
を$X$の期待値という．
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% 基本事項A2.1.1の側注：事象と確率（One More）
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% 側注1
例：1つのさいころを投げる試行において，2の倍数が出る事象を$A$とすると，全事象$U$は，
\[
U=\{1,2,3,4,5,6\}
\]
事象$A$は，
\[
A=\{2,4,6\}
\]
根元事象は，
\[
\{1\},\{2\},\{3\},\{4\},\{5\},\{6\}
\]
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% 基本事項A2.1.2の側注：確率の基本性質（One More）
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% 側注1
$P$は確率（probability）の頭文字である．
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% 基本事項A2.1.3の側注：積事象と和事象，排反事象（One More）
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% 側注1
なお，$\varnothing$はノルウェー語由来の記号である．空集合を表す$\varnothing$や$\emptyset$といった記号は，$\phi$（ファイ）で代用されることもある．
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% 基本事項A2.1.4の側注：余事象の確率（One More）
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% 側注1
事象$A$の起こる確率$P(A)$を計算するとき，「$A$が起こらない」という事象の確率を考える方が楽な場合がある．
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% 基本事項A2.2.1の側注：独立な試行とその確率（One More）
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% 側注1
3つ以上の試行についても同様である．
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% 基本事項A2.2.2の側注：反復試行の確率（One More）
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% 側注1
独立重複試行や，単に重複試行，独立試行ともいう．
% 側注2
$A$が起こる確率が$p$であるから，$A$が起こらない確率は，$1-p$
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% 基本事項A2.2.3の側注：条件付き確率（One More）
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% 側注1
$P(B|A)$と表されることもある．
% 側注2
$n(A)\neq 0,P(A)\neq 0$
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% 基本事項A2.2.4の側注：期待値（One More）
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% 側注1
$p_1+p_2+p_3+\cdots+p_n=1$が成り立つ．
