Suppose that $B _ { 1 }$ and $B _ { 2 }$ are mutually exclusive and complementary events, such that $P \left( B _ { 1 } \right) = .6$ and $P \left( B _ { 2 } \right) = .4$. Consider another event $A$ such that $P \left( A \mid B _ { 1 } \right) = .2$ and $P \left( A \mid B _ { 2 } \right) = .5$. Find $P \left( B _ { 1 } \mid A \right)$.
A) .240
B) .375
C) .625
D) .800

Question

Quizplus · Accepted Answer

Using Bayes' theorem: $P(B_1 \mid A) = \frac{P(A \mid B_1)P(B_1)}{P(A)}$. First, find $P(A)$ using the law of total probability: $P(A) = P(A \mid B_1)P(B_1) + P(A \mid B_2)P(B_2) = 0.2 \times 0.6 + 0.5 \times 0.4 = 0.12 + 0.2 = 0.32$. Then, $P(B_1 \mid A) = \frac{0.2 \times 0.6}{0.32} = \frac{0.12}{0.32} = 0.375$.

Suppose That B1B _ { 1 }B1​ And B2B _ { 2 }B2​

Suppose That $B _ { 1 }$ And $B _ { 2 }$