Q. 4.27 - A First Course in Probability

Question

An insurance company writes a policy to the effect that an amount of money $A$ must be paid if some event $E$ occurs within a year. If the company estimates that $E$ will occur within a year with probability $p$, what should it charge the customer in order that its expected profit will be 10 percent of $A$ ?

Step-by-Step Solution

Answer

The insurance company should charge the customer an amount of $A(p+0.1)$.

Step 1: Given information

An insurance company writes a policy to the effect that an amount of money $A$ must be paid if some event $E$ occurs within a year.

Step 2: Explanation

Let $X$ be a random variable that represents the amount to be paid to the customer.

$X=\left\{\begin{array}{l}0,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }E\text{ does not occur }\\ A,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }E\text{ occurs }\end{array}\right.$

Since $P(X=0)=P\left(E^c\right)=1-p$ and $P(X=A)=P\left(E\right)=p$, the distribution of $X$ is

$X~\left(\begin{array}{cc}0& A\\ 1-p& p\end{array}\right)$

and its expectation is

$E\left[X\right]=(1-p) 0+p A=p A$

Step 3: Final answer

Let's say that the company charges the customer an amount of money $B$. The profit is then $B-X$, so expected profit is $E[B-X]=0.1 A$. By using additivity of expectation we have

$E[B-X]=B-E\left[X\right]=B-p A=0.1 A$

which gives

$B=A(p+0.1)$