Bayes Theorem Venn diagram

S = "smokers in the district",

E1 = "prefer brand A",

E2 = "prefer brand B" and

E is the event "female".

`P(E_1|E)=(P(E_1nnnE))/(P(E_1nnnE)+P(E_2nnnE))`

`=(12/100)/(12/100+24/100)`

`=1/3`

Or equivalently:

`P(E_1|E) = (P(E_1)xxP(E|E_1))/(P(E_1)xxP(E|E_1)+P(E_2)xxP(E|E_2))`

`=(0.4xx0.3)/(0.4xx0.3+0.6xx0.4)`

`=1/3`