Question

Suppose that B1 and B2 are mutually exclusive and complementary events, such that P(B1 ) = .6 and P(B2) = .4. Consider another event A such that P(A | B1) = .2 and P(A | B2) = .5. Find P(A).

Answer 1

Answer:

So, we get that is P(A)=0.32.

Step-by-step explanation:

We know that:

$P(B_1)=0.6\\\\P(B_2)=0.4\\\\P(A|B_1)=0.2\\\\P(A|B_2)=0.5$

We have the formula for probability:

$P(A|B)=\frac{P(A\cap B)}{P(B)}\\\\\implies P(A\cap B)=P(A|B)\cdot P(B)$

So, we calculate:

$P(A\cap B_1)=P(A|B_1)\cdot P(B_1)\\\\P(A\cap B_1)=0.2\cdot 0.6=0.12\\\\\\P(A\cap B_2)=P(A|B_2)\cdot P(B_2)\\\\P(A\cap B_2)=0.5\cdot 0.4=0.2$

We calculate:

$P(A)=P((A\cap B_1)\cup(A\cap B_2))\\\\P(A)=P(A\cap B_1)+P(A\cap B_2)\\\\P(A)=0.12+0.2\\\\P(A)=0.32$

So, we get that is P(A)=0.32.