Favorable and Unfavorable Expected Monetary Values

A person who buys a $1 ticket in the lottery has 1 chance in 20 million of a net gain of $9,999,999 and 19,999,999 chances in 20 million of a net loss of a dollar. So the expected monetary value of this wager equals:

(1/20,000,000 × $9,999,999) (19,999,999/20,000,000 × $1)

That comes out to $0.50. What does this mean? One way of looking at it is as follows: If you could somehow buy up all the lottery tickets and thus ensure that you would win, your $20 million investment would net you $10 million, or $0.50 on the dollar certainly a bad investment. Another way of looking at the situation is as follows: If you invested a great deal of money in the lottery over many millions of years, you could expect to win eventually, but, in the long run, you would be losing fifty cents on every ticket you bought. One last way of looking at the situation is this: You go down to your local drugstore and buy a blank lottery ticket for $0.50. Since it is blank, you have no chance of win- ning, with the result that you lose $0.50 every time you bet. Although almost no one looks at the matter in this way, this is, in effect, what you are doing over the long run when you buy lottery tickets.

We are now in a position to distinguish favorable and unfavorable expected monetary values. The expected monetary value is favorable when it is greater than zero. Changing our example, suppose the chances of hitting a $20 million payoff on a $1 bet are 1 in 10 million. In this case, the state still has the $1 you paid for the ticket, so your gain is actually $19,999,999. The expected monetary value is calculated as follows:

(1/10,000,000 × $19,999,999) (9,999,999/10,000,000 × $1)

That comes to $1. Financially, this is a good bet; for in the long run you will gain $1 for every $1 you bet in such a lottery.

The rule, then, has three parts: (1) If the expected monetary value of the bet is more than zero, then the expected monetary value is favorable. (2) If the expected monetary value of the bet is less than zero, then the expected monetary value is unfavorable. (3) If the expected monetary value of the bet is zero, then the bet is neutral—a waste of time as far as money is concerned.