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Where
The expected value is:
, - ( ) ( ) ( ) ( ) ( )
This means that if one were to play time-after-time, taking into consideration the small
likelihood of winning occasionally, one would be expected to lose, on average, $0.67 per
ticket.
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Notice that we represented the outcomes by using a table, in which we listed the outcomes,
or the individual along with the probability that this occurs, ( ). This is one way in
which to display a probability distribution, or how all probabilities are distributed among
the various outcomes.
Example 3: A fair, six-sided die is tossed repeatedly. The number of dots that are facing
up after each throw is recorded. Define the random variable, find its probability
distribution, and find and interpret the expected value of the random variable.
SOLUTION: We define the random variable,
The different values that can take on are , since we know there are six
sides. Since this is a fair die, each of these six outcomes has an equally likely chance of
appearing, so ( ) , for all values, of . Our probability distribution is thus,
1 2 3 4 5 6
( ) 1/6 1/6 1/6 1/6 1/6 1/6
The expected value is the sum of the products of each outcome value and its associated
probability.
Statistics for Decision-Making in Business © Milos Podmanik Page 139