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EXPECTATION Expectation
is the average of the values you would get in
conducting an experiment or trial exactly the same way many times. In this discussion of expectation, we will
consider two types. One is a numerical expectation and the other is a
mathematical expectation. Numerical Expectation If you tossed a coin 50 times, you would expect the coin
to fall heads (on the average) about 25 times. Your assumption is explained by
the following definition of numerical expectation: If
the probability of success in one trial is p, and k is the total number of
trials, then kp is the expected number of successes in the k trials. In the above example of tossing the coin 50 times, the probability
of heads (successes) is
where
Substituting values in the equation, we find that
= 25 EXAMPLE:
A die is rolled by a player. What is the
expectation of rolling a 6 in 30 trials? SOLUTION:
The probability of rolling a 6 in 1 trial is
and the number of rolls is k=30 therefore,
In words, the player would expect (on the average) to roll
a 6 five times in 30 rolls. Mathematical Expectation We will define mathematical expectation as follows: If, in the event of a successful result, amount a, is
to be received and the probability of success of that event is p, then ap is
the mathematical expectation. If you were to buy 1 of 500 raffle tickets for a video
recorder worth $325.00, what would be your mathematical expectation? In this
case, the product of the amount you stand to win and the probability of winning
is
where a =
amount you stand to win p = probability of success and
Then, by substitution
Thus, you would not want to pay more than 65 cents for the
ticket, unless, of course the raffle were for a worthy cause. EXAMPLE:
To entice the public to invest in their
development, Sunshine Condominiums has offered a prize of $2,000 to 1 randomly
selected family out of the first 1,000 families that participate in the condominium's
tour. 1. What would be each family's mathematical expectation? 2. Would it be worthwhile for the Jones family to spend
$3.00 in gasoline to drive to Sunshine Condominiums to take the tour? SOLUTION: 1.
2. No; since $3.00 is $1.00 over their expectation of
$2.00, it would not be worthwhile for the Jones family to take the tour. PRACTICE PROBLEMS: 1. A box contains 7 slips of paper, each numbered
differently. A girl makes a total of 50 draws, returning the drawn slip after
each draw. a. What is the probability of drawing a selected numbered
slip in 1 drawing? b. How many times would the girl expect to draw the single
selected numbered slip in the 50 draws? 2. In a winnertakeall tournament among four professional
tennis players, the prize money is $500,000. Joe Conners, one of the tennis
players, figures his probability of winning is 0.20. a. What is his mathematical expectation? b. Would he be better off if he made a secret agreement
with the other tennis players to divide the prize money evenly regardless of
who wins? ANSWERS:
2. a. $100,000 b. Yes; he would be better off, since he would make
$125,000, which is greater than his expectation of $100,000. 