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EXPECTED VALUE
The expected value of a variable is the long-run average value of the variable.
Expected value can also be viewed as the average value of a statistic over an infi-
nite number of samples from the same population. 
Studies of expected value emerged from problems in gambling. How much
is a lottery ticket worth? Consider a lottery run by a service organization: a thou-
sand tickets are offered at a dollar each; first prize is $500; there are two second
prizes of $100; and the remaining income from ticket sales is designated for char-
ity. There are three probabilities in this problem: The probability of having the
first-prize ticket is 1 out of 1,000, or 0.001; the probability of a second place
ticket is 0.002; and the probability of winning nothing is 0.997. The average of
prizes weighted with corresponding probabilities gives the expected winning for
a ticket: 
500 • 0.001 + 100 • 0.002 + 0 • 0.997 = 0.700. The expected-prize
value for one of these lottery tickets is $0.70. Since the ticket costs a dollar, the
expected loss on a ticket is $0.30. For this model to hold, one must assume that
a ticket would be purchased from many such lotteries. This assumption is met by
state lotteries that sell millions of tickets or Las Vegas slot machines, which are
played millions of times each day. Neither lotteries nor slot machines are fair
games. The expected net winning for each ticket in the lottery or each play of a
slot machine is a negative number. This indicates that these games of chance rep-
resent a long-term loss for the regular gambler.
The concept of weighting costs by probabilities is used in finance, investing,
insurance, industrial decision-making, and law to determine expected values.
Bankers and investors use several indicators based on expected value. One
example is expected return, an expected value on a risky asset based on the prob-
ability distribution of possible rates of return that might include U.S. Treasury
notes, stock-market indices, and a risk premium. Industrial decision-making uses
expected values to compute projected costs of different options. For example, an
oil company may hold property that it may choose for oil drilling, hold for later
drilling, or sell. Each of these options is associated with costs. The company can