Contoh Soal 3 Peluang Kejadian Majmuk

$\begin{array}{ll}\ 11.&\textrm{Jika kejadian}\: \: A\: \: \textrm{dan}\: \: B\: \: \textrm{adalah dua kejadian}\\ &\textrm{dengan}\: \: P(A)=\displaystyle \frac{8}{15},\: P(B)=\displaystyle \frac{7}{12},\: \: \textrm{dan}\\ &P(A| B)=\displaystyle \frac{4}{7},\: \textrm{maka nilai}\: \: P(B|A)=\: ....\\ &\begin{array}{llllllll}\\ \textrm{a}.&\displaystyle \frac{8}{45}&&&\textrm{d}.&\color{red}\displaystyle \frac{5}{8}\\\\ \textrm{b}.&\displaystyle \frac{1}{3}&\textrm{c}.&\displaystyle \frac{3}{8}&\textrm{e}.&\displaystyle \frac{7}{15} \end{array}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Kejadian di atas adalah contoh}\\ &\textrm{kejadian}\: \: \textbf{tidak saling bebas (bersyarat)}.\\ &\color{red}\textrm{Diketahui bahwa}\\ &P(A)=\displaystyle \frac{8}{15},\: P(B)=\displaystyle \frac{7}{12},\: P(A|B)=\displaystyle \frac{4}{7}\\ &\color{red}\textrm{Ditanyakan nilai}\: \: P(B|A)=\: ...?\\ &\color{purple}\textrm{maka}\\ &\begin{aligned}\color{blue}P(A\cap B)&=\color{blue}P(A\cap B)\\ P(A)\times P(B|A)&=P(B)\times P(A|B)\\ P(B|A)&=\displaystyle \frac{P(B)\times P(A|B)}{P(A)}\\ &=\displaystyle \frac{\left ( \displaystyle \frac{7}{12} \right )\times \left ( \displaystyle \frac{4}{7} \right )}{\displaystyle \frac{8}{15}}\\ &=\displaystyle \frac{\displaystyle \frac{1}{3}}{\displaystyle \frac{8}{15}}\\ &=\displaystyle \frac{1}{3}\times \frac{15}{8}\\ &=\color{red}\displaystyle \frac{5}{8} \end{aligned} \end{aligned} \end{array}$