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 localid="1646823185045" $p_f,$where $p_f\ne p_m$. 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 the year $i,$show that

$P\left(A_2\mid A_1\right)>P\left(A_1\right)$

Give an intuitive explanation of why the preceding inequality is true.

From the given information, 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 localid="1646823204596" $p_f.$

Here,$p_f\ne p_m$

Let $M$and $F$denote the events that the policyholder is male and that the policyholder is female respectively. According to the Condition the following results are shown below.

$P\left(A_2\mid A_1\right)=\frac{P\left(A_1{A}_2\right)}{P\left(A_1\right)}$

$=\frac{P\left(A_1{A}_2\mid M\right)\alpha +P\left(A_1{A}_2\mid F\right)(1-\alpha )}{P\left(A_1\mid M\right)\alpha +P\left(A_1\mid F\right)(1-\alpha )}$

We get,

$=\frac{p_m^2\alpha +p_f^2(1-\alpha )}{p_m\alpha +p_f(1-\alpha )}$

If $\frac{p_m^2\alpha +p_f^2(1-\alpha )}{p_m\alpha +p_f(1-\alpha )}>p_m\alpha +p_f(1-\alpha )$

So being indebted will work with the second equality and prove its validity to prove our problem.

Show that ${p^2}_m\alpha +{p_f}^2(1-\alpha )>{\left[p_m\alpha +p_f(1-\alpha )\right]}^2$Simplifying the equation, that is,

$p^2{}_m\alpha +p_f^2(1-\alpha )>\left[p_m{\alpha }^2+p_f^2(1-\alpha {)}^2+2\alpha (1-\alpha )p_m{p}_f\right]$

$\Rightarrow p^2{}_m\left(\alpha -{\alpha }^2\right)+p^2,\left(\alpha -{\alpha }^2\right)>\left[2\left(\alpha -{\alpha }^2\right)p_m{p}_f\right]$

Simplifying the equation,

$\Rightarrow p^2{}_m+p^2{}_f>2p_m{p}_f$

$\Rightarrow p_m^2+p^2{}_f-2p_m{p}_f>0$

We get,

$\Rightarrow {\left(p_m-p_f\right)}^2>0\left(\text{Since}{p}_m\ne p_f\right)$

The inequality follows. Because from the provided data, the policyholder had a claim in the year $1.$And it creates more likely that was a kind of policyholder having a larger claim probability.