Mentor: Every day, the stock of Auto Superior, Inc., has 1 chance out of 3 to make a payoff. The stock of Books Unlimited, Inc., has 1 chance out of 5 to make payoff. If the stocks cost the same, which one will bring in more money over time?
Student 1: One-third is more than one-fifth, so the first stock is better.
Student 2: Wait! We do not have enough information to decide which stock is better. We need to know how large the payoffs are.
Mentor: Here is some additional information: if the stocks make payoffs, the stock of Auto Superior, Inc. (we will call it stock A for short) will pay 2 points, and stock of Books Unlimited, Inc. (stock B) will pay 4 points. Which one would you buy, knowing that? Why?
Student 2: Stock B is better than stock A.
Mentor: If you watch the stocks for one hundred days, how much do you expect them to pay per day, on the average?
Student 1: Stock A will pay on approximately one-third of all days, which means 1/3*100. Each time, it will pay 2 points, making the total money expected 1/3*100*2. To find the average, we have to divide by 100 (for the number of days), so the answer is:
Student 2: We can skip multiplying by 100 and then dividing by 100, so we will get 1/3*2=2/3.
Student 3: For stock B, we have the expected average of 1/5*4=4/5. I have skipped the manipulations with 100.
Student 1: That's how we can always do it. To find the average expected payoff or winning, we multiply the probability of winning by the number of points won each time.
Student 2: Aha! We can expect stock B to pay more than stock A on the average!
Mentor: Can you change the stocks' payoffs to make the stocks pay equally well over long periods of time?
Student: If we make the first one pay 30 points, and the second one 50 points, then their average payoffs will be the same (10 points per day).
Mentor: By the way, in mathematics the average payoff is sometimes called expected value of payoff. Can you think of more examples where the chances of winning are different, but the expected values are the same? It might also be interesting to start with payoffs and find the probabilities of winning that will make the game fair (that is, to make the expected values of winning the same).