Using Probability to Make Decisions Under Uncertainty

Introduction: Decision-Making Under Uncertainty

Businesses are often faced with the need to make decisions under conditions of uncertainty. There are different approaches which decision makers may use. In some cases there may be the use of intuition, but even where it is believed that a decision is based on intuition or a "gut" feeling, there will be a potential range of influences — such as previous experience, assumptions, and stereotypes (Ames et al., 2012). However, in most cases, especially where a decision will impact the bottom line, managers will seek to base decisions on more than simply feelings or potentially subjective assessments.

One approach is the utilization of probability — assessing the likelihood that a particular outcome will or will not happen (Anderson et al., 2001). To illustrate how probability may be used to assess a decision under uncertainty, consider the following example. A farmer (or other trader) has been offered a stall at an annual festival. The trader knows that if the weekend remains sunny, the festival will attract a high number of attendees and there is the potential for a high profit. However, if there is rain, attendance will drop and profit potential will fall accordingly. The trader has also learned that one of their main competitors has been invited; if that competitor does not attend, the trader stands to earn a greater profit than if competition is present.

This is a scenario involving different variables but only a fixed number of potential outcomes — a situation well suited to probabilistic assessment.

The two variables — weather and the presence of competition — are independent of each other. In a classic approach to probability, a decision tree may be used to assess the different potential outcomes. The decision tree (Figure 1) maps each consideration in sequence.

Building the Decision Tree

Figure 1: Decision Tree

The decision tree reveals four potential outcomes, which may be described as FN (fine weather, no competition), FC (fine weather, competition present), RN (rain, no competition), and RC (rain, competition present). In the classic approach there is a starting assumption that the different outcomes are all equally likely (Stine and Foster, 2010). This assumption holds for many textbook examples, such as coin tossing — where heads and tails are equally likely — or rolling an unloaded die. If this assumption were accurate, there would be a 0.5 probability of fine weather and a 0.5 probability of rain. Similarly, there would be a 0.5 probability of no competition and a 0.5 probability of competition being present. The overall probability of each outcome would then be calculated by multiplying the probabilities at each stage of the decision tree: 0.5 × 0.5 = 0.25.

Estimating Probabilities for Each Variable

In practice, however, the business manager is aware that an even probability distribution is unlikely to be accurate. It is therefore necessary to consider different approaches for estimating probability more reliably.

Historical data can be useful for assessing the weather variable — specifically, by examining the proportion of times that particular weekend has experienced rain in past years. If, for example, the weather was fine in six out of the last ten years and it rained in four out of ten, it may be estimated that the probability of fine weather is 0.6 and the probability of rain is 0.4. Historical data is a practical starting point for forecasting, though past patterns are not necessarily reliable predictors of future events.

To obtain a more reliable assessment, the trader could consult a weather service, which produces forecasts based on large volumes of meteorological data. However, doing so comes at a cost — both financial and temporal. If a fast decision is needed, the manager may need to rely on information already at hand, trading accuracy for speed. If time permits, the added accuracy may well be worth the wait and expense. If the weather service forecast indicates only a 30% chance of rain over that weekend, this figure can be incorporated directly into the probability calculation, yielding a 0.7 probability of fine weather and a 0.3 probability of rain.

Assessing the probability of competition is more difficult. The trader knows that only one competitor has been invited. It is also known that in past years the competitor has attended this festival in some years and an alternate festival in others — five years at each. As no more accurate method is readily available, the most viable approach is to rely on this historical data, assigning a 0.5 probability to each outcome (competition present or absent).

Conclusion and Recommendation

From the revenue table it is apparent that there is only one outcome that produces a loss: the scenario in which it rains and competition is present. This RC outcome carries a probability of only 0.15, meaning there is an 0.85 probability that the trader will make a profit. The use of probability theory in this way allows a business manager to move beyond gut feeling and subjective judgment toward a structured, evidence-based assessment of risk and reward.