Joe sits staring into his beer as his favorite baseball team, the Chicago Cubs, loses another game. The Cubbies have some good young players, so let’s ask Joe, “What’s the chance that the Cubs will go to the World Series next year?” Joe brightens up. “Oh, about 10%,” he says.

Does Joe assign probability 0.10 to the Cubs’ appearing in the World Series? The outcome of next year’s pennant race is certainly unpredictable, but we can’t reasonably ask what would happen in many repetitions. Next year’s baseball season will happen only once and will differ from all other seasons in players, weather, and many other ways. The answer to our question seems clear: if probability measures “what would happen if we did this many times,” Joe’s 0.10 is not a probability. Probability is based on data about many repetitions of the same random phenomenon. Joe is giving us something else, his personal judgment.

Yet we often use the term “probability” in a way that includes personal judgments of how likely it is that some event will happen. We make decisions based on these judgments—we take the bus downtown because we think the probability of finding a parking spot is low. More serious decisions also take judgments about likelihood into account. A company deciding whether to build a new plant must judge how likely it is that there will be high demand for its products three years from now when the plant is ready. Many companies express “How likely is it?” judgments as numbers—probabilities—and use these numbers in their calculations. High demand in three years, like the Cubs’ winning next year’s pennant, is a one-time event that doesn’t fit the “do it many times” way of thinking. What is more, several company officers may give several different probabilities, reflecting differences in their individual judgment. We need another kind of probability, personal probability.

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Personal probabilities have the great advantage that they aren’t limited to repeatable settings. They are useful because we base decisions on them: “I think the probability that the Patriots will win the Super Bowl is 0.75, so I’m going to bet on the game.” Just remember that personal probabilities are different in kind from probabilities as “proportions in many repetitions.” Because they express individual opinion, they can’t be said to be right or wrong.

This is true even in a “many repetitions” setting. If Craig has a gut feeling that the probability of a head on the next toss of this coin is 0.7, that’s what Craig thinks and that’s all there is to it. Tossing the coin many times may show that the proportion of heads is very close to 0.5, but that’s another matter. There is no reason a person’s degree of confidence in the outcome of one try must agree with the results of many tries. We stress this because it is common to say that “personal probability” and “what happens in many trials” are somehow two interpretations of the same idea. In fact, they are quite different ideas.

Why do we even use the word “probability” for personal opinions? There are two good reasons. First, we usually do base our personal opinions on data from many trials when we have such data. Data from Buffon, Pearson, and Kerrich (Example 2) and perhaps from our own experience convince us that coins come up heads very close to half the time in many tosses. When we say that a coin has probability one-half of coming up heads on this toss, we are applying to a single toss a measure of the chance of a head based on what would happen in a long series of tosses. Second, personal probability and probability as long-term proportion both obey the same mathematical rules. Both kinds of probabilities are numbers between 0 and 1. We will look at more of the rules of probability in the next chapter. These rules apply to both kinds of probability.

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Although “personal probability” and “what happens in many trials” are different ideas, what happens in many trials often causes us to revise our personal probability of an event. If Craig has a gut feeling that the probability of a head when he tosses a particular coin is 0.7, that’s what Craig thinks. If he tosses it 20 times and gets nine heads, he may continue to believe that the probability of heads is 0.7—because personal probabilities need not agree with the results of many trials. But he may also decide to revise his personal probability downward based on what he has observed. Is there a sensible way to do this, or is this also just a matter of personal opinion?

In statistics, there are formal methods for using data to adjust personal probabilities. These are called Bayes’s procedures. The basic rule, called Bayes’s theorem, is attributed to the Reverend Thomas Bayes, who discussed the rule in “An Essay towards Solving a Problem in the Doctrine of Chances” published in 1764. The mathematics is somewhat complicated, and we will not discuss the details. However, the use of Bayes’s procedures is becoming increasingly common among practitioners.