2 6 4C H A P T E R 1 2 ■ C h o i c e syour net gain on your $1 ticket is only $9,999,999. The state, after all, still has
your original $1. So the net gain equals the payoff minus the cost of betting.
This is not something that those who win huge lotteries worry about, but
taking into account the cost of betting becomes important when this cost
becomes high relative to the size of the payoff. There is nothing complicated
about the net amount that you lose when you lose on a $1 ticket: It is $1.^1
We can now compute the expected monetary value or financial worth of a
bet in the following way:
Expected monetary value 5
(the probability of winning times the net gain in money of winning) minus
(the probability of losing times the net loss in money of losing)
In our example, 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 3 $9,999,999) 2 (19,999,999/20,000,000 3 $1)
That comes out to 2 $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
financially 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 3 $19,999,999) 2 (9,999,999/10,000,000 3 $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.97364_ch12_ptg01_263-272.indd 264 15/11/13 11:00 AM
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