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.
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
Substituting values in the equation, we find that
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
In words, the player would expect (on the average) to roll a 6 five times in 30 rolls.
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
a = amount you stand to win
p = probability of success
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?
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.
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 winner-take-all 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?
b. Yes; he would be better off, since he would make $125,000, which is greater than his expectation of $100,000.