Answer to Question #209551 in Statistics and Probability for Kyle

Question

Assume that A and B are events, with P(A)= 0.6, P (B|A)= 0.2 and P(B|A c)= 0.1. Find the probability for

a. P(B)

Answer 1

$$P(B|A)=\dfrac{P(A\cap B)}{P(A)}$$

$$=>P(A\cap B)=P(B|A)P(A)$$

$$P(B|A^C)=\dfrac{P(A^C\cap B)}{P(A^C)}$$

$$=>P(A^C\cap B)=P(B|A^C)P(A^C)$$

$$P(A^C)=1-P(A)$$

$$P(B)=P(A\cap B)+P(A^C\cap B)$$

$$=P(B|A)P(A)+P(B|A^C)(1-P(A))$$

$$P(B)=0.2(0.6)+0.1(1-0.6)=0.16$$