Question

P(A1) = 0.60, P(A2) = 0.40, P(B | A1) = 0.05, and P(B | A2) = 0.10. Use Bayes’ theorem to determine P(A1 | B).

Answer 1

The result follows from manipulating the conditional probability. By definition,

`P(A_1\mid B)=(P(A_1\cap B))/(P(B))=(P(B\mid A_1)P(A_1))/(P(B))`

`P(A_1\mid B)=(0.05\cdot0.60)/(P(B))`

By the law of total probability,

`P(B)=P(B\cap A_1)+P(B\cap A_2)=P(B\mid A_1)P(A_1)+P(B\mid A_2)P(A_2)`

So we have

`P(A_1\mid B)=(0.05\cdot0.60)/(0.05\cdot0.60+0.10\cdot0.40)\approx0.4286`