Determining Probability

Determining Probability

How do we determine probability? There are basically two ways: computation and frequency-and-experience. When a model specifies the possible outcomes and their probabilities, we can calculate probabilities by arithmetic computation. When we don’t have all the facts, we estimate probability based on frequency and experience. Frequency is how often an event has occurred in the past; experience is what happened during each event. A simple example: If I drop ten lightbulbs, one at a time, on the floor and each one breaks, what’s the probability that, if I drop an eleventh bulb, it too will break? The observations suggest a high probability, but ten trials alone do not establish a probability of almost 100 percent. How did I reach that conclusion? Frequency—I repeated the event ten times. Experience— the bulbs broke every time.

Relevant information about an event’s circumstances can help us estimate its probability. But what if we know little or nothing about the circumstances? What if someone asked you to estimate which of three political parties will win a coming election in Tanganyika, Africa? Unless by chance you are a student of African affairs, you won’t have the slightest idea. A French nobleman, Marquis Pierre-Simon de Laplace (1749-1827), gave us sound advice about how to deal with such a situation. According to Laplace, if we’re trying to determine which of two or more outcomes will occur, but we don’t have reliable evidence to judge which is more likely, we should assume the probability is equal for all outcomes. This is a modeling assumption, not a deduction from ignorance: different ways of defining the possible outcomes can produce different equal-probability assignments.

Fortunately, we rarely have occasion to resort to Laplace. In almost every problem we face, we recognize similarities with past events of which we have some knowledge, and we use that knowledge as the basis for making probability judgments (estimating probabilities). However, our probability estimates are as vulnerable as any other workings of the human mind to those troublesome mental traits I’ve been talking about. “Similarities with past events” are often among those nonexistent patterns our minds are prone to perceive. And as we are prone to value evidence that supports a favored outcome, so are we prone to believe that a favored outcome is more likely (has a greater probability of occurring) than one unfavored. And we are likely to hold fast to this belief in the face of contrary evidence. Yes, indeed, our minds can as easily mislead our probability estimates as they can any other analytic judgment.

A memorably poignant example of this tendency to inflate the probability of a desired outcome was revealed in a statement by a British Royal Navy commando who took part in the Normandy invasion in World War II. “I don’t think any of us realized the danger we were in. Nobody ever went to Normandy to die; they went to fight and win a war. I don’t think any man that died thought that it was him that was going to be killed.” Inflating or deflating probabilities to conform to our desires is the curse of wishful thinking: it won’t happen to me because I don’t want it to happen. As Francis Bacon said, we prefer to believe what we prefer to be true.

Given the absolutely crucial role of probability in the analysis of problems, we must take great care in the determination of probability assessments.