## Expected Value

expected value The expected value is the anticipated realization from a given payoff matrix and probability

Antidpated rediz^im distribution. It is the weighted-average payoff, where the weights are defined by the probability distribution.

To continue with the previous example, assume that forecasts based on the current trend in economic indicators suggest a 2 in 10 chance of recession, a 6 in 10 chance of a normal economy, and a 2 in 10 chance of a boom. As probabilities, the probability of recession is 0.2, or 20 percent; the probability of normal economic activity is 0.6, or 60 percent; and the probability of a boom is 0.2, or 20 percent. These probabilities add up to 1.0 (0.2 + 0.6 + 0.2 = 1.0), or 100 percent, and thereby form a complete probability distribution, as shown in Table 14.3.

If each possible outcome is multiplied by its probability and then summed, the weighted average outcomes is determined. In this calculation, the weights are the probabilities of occurrence, and the weighted average is called the expected outcome. Column 4 of Table 14.3 illustrates the calculation of expected profits for projects A and B. Each possible profit level in column 3 is multiplied by its probability of occurrence from column 2 to obtain weighted values of the possible profits. Summing column 4 of the table for each project gives a weighted average of profits under various states of the economy. This weighted average is the expected profit from the project.

The expected-profit calculation is expressed by the equation n

Here, ni is the profit level associated with the ith outcome, pi is the probability that outcome i will occur, and n is the number of possible outcomes or states of nature. Thus, E(n) is a weighted average of possible outcomes (the ni values), with each outcome's weight equal to its probability of occurrence.

The expected profit for project A is obtained as follows: 