Probability to Make a Decision Under Uncertainty
Businesses are often faced with the need to make decisions under conditions of uncertainty. There are different approaches which the decision makers may use. In some cases there may be the use of intuition, but even where it is believed that the decision is based on intuition of 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 on the bottom line, managers will seek to make the decision based on more than simply feelings or potentially subjective assessments. One approach is the utilization of probability; assessing the probability that that a particular outcome will or will not, happen (Anderson et al., 2001). To consider the way probability may be used to assess a decision under uncertainty an example may be used. 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 have a high number of attendees and there is the potential for a high profit. However, if there is rain the number of festival attendees will drop and the potential for profits will drop. The trader has also found out that one of their main competitors has been asked to attend, if there is no competition there will be a greater potential level of profit, compared to if competition is present.
This is a scenario where there are different variables, but there are only a fixed number of potential outcomes. This situation is one which is suitable to be assessed with probability. The two variables 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 is shown in figure 1, where the different considerations are made one by one.
Figure 1; Decision Tree
The decision tree shows there are four potential outcomes, which may be described as FN, FC, RN and RC. In the classic approach there is a starting point where there is assumption that the different outcomes are all equally likely (Stein and Foster, 2001). This is true for many of the examples used in books, such as coin tossing where heads and tails are known to be equally lily, or the use of a dice which is not loaded. If this is assumption is accurate then there would be a 0.5 chance of the festival being fine and a 0.5 chance of there being rain. In turn there would be the assumption that there would be a 0.5 chance of there being no competition and a 0.5 chance of competition being present. In each case the probability of each outcome will be the probability of each stage of the decision tree multiplied, so it will be 0.5 x 0.5 = 0.25.
However, the business manager is aware that the probability is unlikely to be even. Therefore it is necessary to look at different approaches which may be used to assess probability. There are a few tools which may be used. Historical data may be useful, looking at the probability that that particular weekend will have rain by assessing the proportion of times the same weekend has had rain in the past. If it is found that in six out if the last ten years the weather was fine, but in four out of the last ten years it rain, it may be extrapolated that the chances of the weather raining fine are 0.6, and the chances of rain is 0.4. Historical data may be used for forecasting, but past patterns will not necessarily be reliable.
To gain a more reliable assessment there are alternatives; the weather service makes advanced forecasts based on a large amount of data. However, to gain this information the business will have to pay a fee. This means that there is a cost associated with gaining the more accurate data. It may also be assumed that if data to be used in the decision is being gained from an external source, there may also be a time delay. If a fast decision is needed, the decision may need to be based only in the information at hand, trading off the potential accuracy to gain the speed. If the decision is not needed quickly, then the extra time may be well spent; accuracy will be gain at the cost of speed. If the forecast from the weather service is that there is only a 30% chance of rain over that weekend, this can be used in the probably calculation, showing a 0.7 probability that it will be fine, and a 0.3 probability it will rain.
Assessing the accurate probability that there will be competition is more difficult. The trader knows that only one competitor has been invited. It may also be known and that in the past some years they have attended the festival, but in other years they have attended an alternate festival; five years at each. As there is no approach that is likely to be accurate, so the most viable approach is to use the historical data, giving 0.5 probability of each potential outcome.
This may then be used to assess the probability of each outcome.
Table 1; Probability table
F/R probability
C/N probability
Overall probability
FN
0.7
0.5
0.35
FC
0.7
0.5
0.35
RN
0.3
0.5
0.15
RC
0.3
0.5
0.15
This shows that the probabilities. However the business will also want to assess the potential financial impact. For this historical data and estimates may be used. If it is assumed that the stall will cost $5,000, and if it is fine net revenues on the goods will be $20,000, but if it is raining the net revenues will only be $8,000. If competition is present the overall level of net revenue will be halved with the trade shared between two traders.
Table 2; Revenues for each outcome
Net revenue
Less fee for stall
Net profit/loss
FN
20,000
5,000
15,000
FC
10,000
5,000
5,000
RN
8,000
5,000
3,000
RC
4,000
5,000
-1,000
From this it is apparent that there is only one outcome where there is a loss; that is where it rains and where the competition is present. This means there is only a 0.15 probability of the firm making a loss and a 0.85 probability that there will be a profit. Therefore, in this case it is recommended that the business takes the stall, as long as there is not a better opportunity available.
References
Ames, D. R; Weber, E. U; Zou, X, (2012), Mind-reading in strategic interaction: The impact of perceived similarity on projection and stereotyping, Organizational Behavior and Human Decision Processes, 117, 96-110
Anderson, David R., Sweeney, Dennis J; Davis, Duane; Utts, Jessica M; Williams, Thomas a; Simon, Marilyn, (2001), Statistics and Research Methods for Managerial Decisions, South-Western College Publishing
Stine, Robert a; Foster, Dean, (2010), Statistics for Business: Decision Making and Analysis, Pearson
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