Subjective Versus Objective Probabilities

We now ask a very basic question: Where do the probabilities in a probability distribu- tion come from? A complete answer to this question could lead to a chapter by itself,

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so we only briefly discuss the issues involved. There are essentially two distinct ways to assess probabilities, objectively and subjectively. Objective probabilities are those that can be estimated from long-run proportions, whereas subjective probabilities can- not be estimated from long-run proportions. Some examples will make this distinction clearer.

Consider throwing two dice and observing the sum of the two sides that face up. What is the probability that the sum of these two sides is 7? You might argue as follows. Because there are ways the two dice can fall, and because exactly 6 of these result in a sum of 7, the probability of a 7 is . This is the equally likely argument we discussed previously. It reduces probability to counting.

What if the dice are weighted in some way? Then the equally likely argument is no longer valid. You can, however, toss the dice many times and record the proportion of tosses that result in a sum of 7. This proportion is called a relative frequency.

6/36 = 1/6

6 * 6 = 36

The relative frequency of an event is the proportion of times the event occurs out of the number of times the random experiment is run. A relative frequency can be recorded as a proportion or a percentage.

A famous result called the law of large numbers states that this relative frequency, in the long run, will get closer and closer to the “true” probability of a 7. This is exactly what we mean by an objective probability. It is a probability that can be estimated as the long- run proportion of times an event occurs in a sequence of many identical experiments.

If you are flipping coins, throwing dice, or spinning roulette wheels, objective proba- bilities are certainly relevant. You don’t need a person’s opinion of the probability that a roulette wheel, say, will end up pointing to a red number; you can simply spin it many times and keep track of the proportion of times it points to a red number. However, there are many situations, particularly in business, that cannot be repeated many times—or even more than once—under identical conditions. In these situations objective probabilities make no sense (and equally likely arguments usually make no sense either), so you must resort to subjec- tive probabilities. A subjective probability is one person’s assessment of the likelihood that a certain event will occur. We assume that the person making the assessment uses all of the information available to make the most rational assessment possible.

This definition of subjective probability implies that one person’s assessment of a prob- ability can differ from another person’s assessment of the same probability. For example, consider the probability that the Indianapolis Colts will win the next Super Bowl. If you ask a casual football observer to assess this probability, you will get one answer, but if you ask a person with a lot of inside information about injuries, team cohesiveness, and so on, you might get a very different answer. Because these probabilities are subjective, people with different information typically assess probabilities in different ways.

Subjective probabilities are usually relevant for unique, one-time situations. However, most situations are not completely unique; you often have some history to guide you. That is, historical relative frequencies can be factored into subjective probabilities. For example, suppose a company is about to market a new product. This product might be quite different in some ways from any products the company has marketed before, but it might also share some features with the company’s previous products. If the company wants to assess the probability that the new product will be a success, it will certainly analyze the unique features of this product and the current state of the market to obtain a subjective assessment. However, the company will also look at its past successes and failures with reasonably similar products. If the proportion of successes with past products was 40%, say, then this value might be a starting point in the assessment of this product’s probability of success.

All of the “given” probabilities in this chapter and later chapters can be placed some- where on the objective-to-subjective continuum, usually closer to the subjective end. An important implication of this placement is that these probabilities are not cast in stone; they are only educated guesses. Therefore, it is always a good idea to run a sensitivity analysis (especially on a spreadsheet, where this is easy to do) to see how any “bottom-line” answers depend on the given probabilities. Sensitivity analysis is especially important in Chapter 6, when we study decision making under uncertainty.

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P R O B L E M S

Note: Student solutions for problems whose numbers appear within a colored box are available for purchase at www.cengagebrain.com.

Level A

1. In a particular suburb, 30% of the households have