독립사건 확률

$\mathrm{P} ( A \cap B ) = 2 \mathrm{P} ( A \cap B ^{C} )$ $\mathrm{P} ( A ) \times \mathrm{P} ( B ) = 2 \mathrm{P} ( A ) \times \mathrm{P} ( B ^{C} )$ (∵$A$,$B$는 독립) $\mathrm{P} ( B ) = 2 \mathrm{P} ( B ^{C} )$ (∵ $\mathrm{P} ( A ) \ne 0$) $\mathrm{P} ( B ) = 2 ( 1 - \mathrm{P} ( B ) )$ $3 \mathrm{P} ( B ) = 2$, $\displaystyle \mathrm{P} ( B ) = \frac{2}{3} \cdots \cdots \text{㉠}$ 또 $\displaystyle \mathrm{P} ( A ^{C} \cap B ) = \frac{1}{12}$, $\displaystyle \mathrm{P} ( A ^{C} ) \times \mathrm{P} ( B ) = \frac{1}{12}$ ㉠에서 $\displaystyle \mathrm{P} ( A ^{C} ) \times \frac{2}{3} = \frac{1}{12}$ $\displaystyle \mathrm{P} ( A ^{C} ) = \frac{1}{8}$ $\displaystyle \therefore P ( A ) = 1 - \mathrm{P} ( A ^{C} ) = 1 - \frac{1}{8} = \frac{7}{8}$