Question

A man won ​$5 million by scoring 16 goals in 23 seconds in a contest at a sports game that was sponsored by a business. Did the business risk the ​$ ​million? No! The business purchased event insurance from a company specializing in promotions at sporting events. The event insurance company estimates the probability of a contestant winning the contest​ and, for a modest​ charge, insures the event. The promoters pay the insurance premium but take on no added risk as the insurance company will make the large payout in the unlikely event that a contestant wins. To see how it​ works, suppose that the insurance company estimates that the probability a contestant would win the contest is 0.0009 and that the insurance company charges ​$9,500

a. Calculate the expected value of the profit made by the insurance company.

Answer 1

In this case, the expected value of profit is the sum of the probability that an event occurs. and the profit or loss associated with the event.
That is,

Where E(x) is the expected value of profit,   is the probability of an event , and is the profit(or loss) associated with that event

There is a 100%(1) probability that the insurance company will get its $9500 dollar premium,which is a profit of $9500

There is a 0.0009(0.000009%) probability that the insurace company will have to pay out(loss) 5 million dollars(since it is a loss, while calculating expected value of profit, we have to take the value as negative 5 million)

There is a 0.9991(1-0.009) probability that the insurace company will have to pay out nothing(no profit or loss)

So, the expected value E(x) for the profit made by the insurance company will be

Hence, the expected value of the profit made by the insurance company is $5000.