Probability theory is a decision-making tool used in risk situations—situations in which decision makers are not completely sure of the outcome of an implemented alternative.³⁶ Probability refers to the likelihood that an event or outcome will actually occur, which is estimated by calculating an expected value for each alternative considered. Specifically, the expected value (EV) for an alternative is the income (I) that the alternative would produce, multiplied by its probability of producing that income (P). In formula form, EV = I × P. Decision makers generally choose and implement the alternative with the highest expected value.³⁷

An example will clarify the relationship among probability, income, and expected value. A manager is trying to decide where to open a store that specializes in renting surfboards. She is considering three possible locations (A, B, and C), all of which seem feasible. For the first year of operation, the manager has projected that, under ideal conditions, her company would earn $90,000 in Location A, $75,000 in Location B, and $60,000 in Location C. After studying historical weather patterns, however, she has determined that there is only a 20 percent chance—or a 0.2 probability—of ideal conditions occurring during the first year of operation in Location A. Locations B and C have a 0.4 and a 0.8 probability, respectively, of ideal conditions during the first year of operations. Expected values for each of these locations are as follows: Location A—$18,000; Location B—$30,000; Location C—$48,000. Figure 6.5 shows the situation this decision maker faces. According to her probability analysis, she should open a store in Location C, the alternative with the highest expected value.

In the previous section, probability theory was applied to a relatively simple decision situation. Some decisions, however, are more complicated and involve a series of steps. These steps are interdependent; that is, each step is influenced by the step that precedes it. A decision tree is a graphic decision-making tool typically used to evaluate decisions involving a series of steps.³⁸

John F. Magee developed a classic illustration that outlines how decision trees can be applied to a production decision.³⁹ In his illustration (see Figure 6.6), the Stygian Chemical Company must decide whether to build a small or a large plant to manufacture a new product with an expected life of 10 years (Decision Point 1 in Figure 6.6). If the choice is to build a large plant, the company could face high or low average product demand, or high initial and then low demand. If, however, the choice is to build a small plant, the company could face either initially high or initially low product demand. If the small plant is built and product demand is high during an initial two-year period, management could then choose whether to expand the plant (Decision Point 2). Whether the decision is made to expand or not to expand, management could then face either high or low product demand.

Now that various possible alternatives have been presented, the financial consequence of each different course of action must be compared. To adequately compare these consequences, management must do the following:

1. Study estimates of investment amounts necessary for building a large plant, for building a small plant, and for expanding a small plant.

2. Weigh the probabilities of facing different product demand levels for various decision alternatives.

3. Consider projected income yields for each decision alternative.

Analysis of the expected values and net expected gain for each decision alternative helps management decide on an appropriate choice.⁴⁰ Net expected gain is defined in this situation as the expected value of an alternative minus the investment cost. For example, if building a large plant yields the higher net expected gain, Stygian management should decide to build the large plant.⁴¹