A fire insurance company has high-risk, medium-risk, and low-risk clients, who have, respectively, probabilities .02, .01, and .0025 of filing claims within a given year. The proportions of the numbers of clients in the three categories are .10, .20, and .70, respectively. What proportion of the claims filed each year come from high-risk clients?
Let the events are
\(\mathrm{C}_{1}-\) Claim from high Risk client
\(\mathrm{C}_{2}-\) Claim from Medium - Risk client.
\(\mathrm{C}_{3}-\) Claim from Low - Risk client
And Let \(A, B, C\) be the Proportions of High- Risk, Medium- Risk , Low -Risk clients respectively. Therefore , Required Probability is
$$ \begin{aligned} P\left(A / C_{1}\right) &=\frac{P\left(C_{1} / A\right) \cdot P(A)}{P\left(C_{1} / A\right) \cdot P(A)+P\left(C_{2} / B\right) \cdot P(B)+P\left(C_{3} / C\right) \cdot P(C)} \\ &=\frac{.02 \times 0.1}{0.02 \times 0.1+0.01 \times 0.2+0.0025 \times 0.7} \\ &=\frac{0.002}{0.002+0.002+0.00175} \\ &=\frac{0.002}{0.00575} \\ &=0.35 \end{aligned} $$