제비뽑기 조건부확률 보기

갑이 당첨될 확률 P$\displaystyle ( A ) = \frac{2}{5}$ 을이 당첨될 확률 ∙ 갑이 당첨됐을 경우 P$\displaystyle ( A \cap B ) = \frac{2}{5} \times \frac{1}{4} = \frac{1}{10}$ ∙ 갑이 당첨되지 않았을 경우 P$\displaystyle ( A ^{C} \cap B ) = \frac{3}{5} \times \frac{2}{4} = \frac{3}{10}$ ∴ P$\displaystyle ( B ) = \frac{1}{10} + \frac{3}{10} = \frac{2}{5}$ ㄱ. 참 $\displaystyle \mathrm{P} ( A ) = \mathrm{P} ( B ) = \frac{2}{5}$ ㄴ. 거짓 $\displaystyle \mathrm{P} ( B | A ) = \frac{\mathrm{P} ( A \cap B )}{\mathrm{P} ( A )} = \frac{\frac{1}{10}}{\frac{2}{5}} = \frac{1}{4}$ $\displaystyle \mathrm{P} ( B | A ^{C} ) = \frac{\mathrm{P} ( A ^{C} \cap B )}{\mathrm{P} ( A ^{C} )} = \frac{\frac{3}{10}}{\frac{3}{5}} = \frac{1}{2}$ ∴ $\mathrm{P} ( B | A ) < \mathrm{P} ( B | A ^{C} )$ ㄷ. 참 $\displaystyle \mathrm{P} ( B | A ) = \frac{\mathrm{P} ( A \cap B )}{\mathrm{P} ( A )}$ $\displaystyle \mathrm{P} ( A | B ) = \frac{\mathrm{P} ( A \cap B )}{\mathrm{P} ( B )}$ $\mathrm{P} ( A ) = \mathrm{P} ( B )$이므로 $\mathrm{P} ( B | A ) = \mathrm{P} ( A | B )$ 그러므로 옳은 것은 ㄱ, ㄷ이다.