91Ó°ÊÓ

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,$p_f≠p_m$

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.

$P\left(A_2âˆ\pounds A_1\right)=\frac{P\left(A_1{A}_2\right)}{P\left(A_1\right)}$

$=\frac{P\left(A_1{A}_2âˆ\pounds M\right)\mathrm{Î\pm }+P\left(A_1{A}_2âˆ\pounds F\right)(1-\mathrm{Î\pm })}{P\left(A_1âˆ\pounds M\right)\mathrm{Î\pm }+P\left(A_1âˆ\pounds F\right)(1-\mathrm{Î\pm })}$

We get,

$=\frac{p_m^2\mathrm{Î\pm }+p_f^2(1-\mathrm{Î\pm })}{p_m\mathrm{Î\pm }+p_f(1-\mathrm{Î\pm })}$

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

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

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

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

$⇒p^2{}_m\left(\mathrm{Î\pm }-{\mathrm{Î\pm }}^2\right)+p^2,\left(\mathrm{Î\pm }-{\mathrm{Î\pm }}^2\right)>\left[2\left(\mathrm{Î\pm }-{\mathrm{Î\pm }}^2\right)p_m{p}_f\right]$

Simplifying the equation,

$⇒p^2{}_m+p^2{}_f>2p_m{p}_f$

$⇒p_m^2+p^2{}_f-2p_m{p}_f>0$

We get,

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

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.