1. Math 409/409G History of Mathematics Expected Value

2. The general description of an expected value problem is that someone, perhaps you, pays to repeatedly “play a game” in the hopes of winning a big payoff. Knowing how to compute the expected value tells you how to determine if, in the long run, you will come out the winner or the loser.

3. For example, if you go to Las Vegas to play the roulette wheel, will you come home a winner or a pauper? You’ll probably come home a loser because the folks in Vegas hirer mathematicians to determine how much they should charge so that you will most likely be the loser so they can pocket your money.

4. Even if you don’t think of yourself as a Vegas style gambler, you’ve played (or will play) the game when you pay for any kind of insurance (life, health, etc.) Any company that offers you any kind of insurance has used the mathematical concept of expected value to insure that, in the long run, they make a profit. Here’s how they do it, described in terms of a game.

5. Expected Value of a Game The a i ’s are the possible outcomes of playing the game. The p i ’s are the probability that a i will occur. If E = 0, the game is called a fair game. In this case, you break even by winning as much as you lose.

6. Example A term life insurance will pay a beneficiary a certain sum of money upon the death of the policyholder. The policies have premiums that must be paid annually. Suppose that a life insurance company sells a $250,000 one-year term life insurance policy to a 20-year-old male for $350. The probability that the male will survive the year is 0.99865. What’s the expected value of this policy to the insurance company?

7. The insurance company must ask the question, “what does the company get or lose if he lives or want if he dies?” So in terms of expected value, the insurance company must compute: E(lives or dies)  E(lives) + E(dies).

8. The probability that he lives is 0.99865, and he paid $350 for the policy. So E(lives)  $350(0.99865). On the other hand, if he dies (probability = 1 - 0.99865 = 0.003135) the company had to pay $250,000. So E(dies)  (- $250,000)(0.003135). So the expected value for the insurance co. is E(lives) + E(dies)  ···  $12.50.

9. This ends the lesson on Expected Value