독립사건 확률

$A$, $B$ 가 서로 독립 $\leftrightarrow$ $A$, $B ^{c}$도 독립 $\displaystyle \therefore \mathrm{P} \mathit{(} A \cup B ) = \mathrm{P} ( A ) + P ( B ) - P ( A ) \cdot P ( B ) = \frac{4}{5}$ $\cdots \cdots$ (1) $\displaystyle \mathrm{P} ( A \cap B ^{c} ) = P ( A ) \cdot P ( B ^{c} ) = P ( A ) \cdot ( 1 - P ( B ) ) = \frac{1}{3}$ $\cdots \cdots$(2) (1)식을 변형하면 $\displaystyle \mathrm{P} ( A ) ( 1 - P ( B ) ) + P ( B ) = \frac{4}{5}$ (2)식을 (1)식에 대입하여 계산하면 $\displaystyle \mathrm{P} ( B ) = \frac{7}{15}$ $\cdots \cdots$(3) (3)식을 (2)식에 대입하면 $\displaystyle \therefore \mathrm{P} ( A ) = \frac{5}{8}$