conditional probability question from sheldon ross In any given year a male automobile policyholder will make a claim with probability $p_{m}$, and a female policyholder will make a claim with probability $p_{f}$, where $p_{f} \neq p_{m}$. The fraction of the policyholders that are male is $\alpha, 0 < \alpha < 1$. A policyholder is randomly chosen. If $A_{i}$ denotes the event that this policyholder will make a claim in year $i$, show that $P(A_{2}\mid A_{1}) > P(A_{1})$. 
It appears that $A_{2}$ and $A_{1}$ are independent events, hence I don't understand why the inequality should hold. Can anybody please help? Thanks a lot.

Intuitively, if a policy holder makes a claim in year $1$ then that event suggests that they are more likely to have been in the group which is more likely to make a claim than that group's share of the general population.  
In turn, being more likely to be to the group which is more likely to make a claim suggests that they are more likely than the general population to make a claim in year $2$.
You could formalise this into conditional probability expressions, but I doubt that doing so would increase understanding by much.
