Probability

How we can – or cannot – predict outcomes

By Steve Humphrey
Illustration by
Andrea Hutchinson

The origins of probability theory can be traced back to gambling (of course). Gamblers approached the French mathematician Blaise Pascal and asked if he could devise a way to improve their success at the card table. Pascal communicated with Pierre de Fermat (of “Fermat’s last theorem” fame) and together they began what is now a quite complex and intricate theory. But we will stay with the most basic stuff.

What they were trying to do is calculate the odds that a certain outcome will occur, given less than perfect information as input. For example, what are the odds that a coin, when flipped, will turn up heads? Or that a die will come up six when rolled? Or what are the chances of being dealt a royal flush in a poker game? In all of these cases, there are only a finite number of possible outcomes, so the calculation is fairly simple. Heads is one of only two possible outcomes, so the odds of getting heads is one over two, or .5. (Probabilities are expressed as numbers between zero and one.)

The chances of a die coming up six is one over six. The poker case is exactly the same, though complicated by the large number of possible hands. There are four ways of getting a royal flush, and there are 2,598,960 possible poker hands. So, the odds are four over that big number, which is 0.00015 – very low, indeed. This way of computing probability is called the classical theory.

But what do we do when there are a much larger, perhaps infinite, number of possible outcomes? In that case, we use what is called the frequency interpretation, and in this case, we have to discover rather than calculate the odds. A good example would be to calculate the fatality rate associated with the COVID-19 virus. In principle, we would simply divide the number of fatalities due to the disease by the number of cases of the virus. You have probably seen a lot of different numbers published. Why? Because we don’t know exactly how many people have contracted the disease. All we have access to is the number of confirmed cases, confirmed by testing.

But there haven’t been enough test kits available, and many people are infected but show no symptoms, so they are not tested. Some now believe that a significant number of people, especially on the West Coast, got the virus and either had no symptoms or recovered from what they thought was a bad cold. In this case, the denominator gets much higher, while the numerator stays roughly the same. (People might have died, but it was attributed to the flu.) In fact, I am fairly convinced that I had it back in late January as a result of teaching a class at University of California, Santa Barbara, which included several students fresh from China.

We can use the frequency interpretation to check our classical calculations. Suppose we want to know if we are rolling a “fair” die: i.e. one that is not biased to yield a particular outcome. We could roll the die a large number of times and count how many times each possible outcome arises. If we get a disproportionate number of threes, we might suspect the die has been tampered with. But what counts as a “large number?” For any finite number, the odds for each outcome will be “close” to 1/6 but will not be exactly 1/6. And, it would be easy to find long stretches of rolls in which the odds are very far from 1/6. But we can’t roll the die an infinite number of times. We can’t even roll it a million times (it would break, or our arm would fall off.)

There are two different ways of providing an interpretation of probability. (Philosophers love to make distinctions. There are two kinds of people in the world: those who divide the world into two kinds of people and those who don’t.) Traditionally, probability is seen as a reflection of ignorance. If we knew all the relevant conditions in the coin flip (air density, force applied to the edge of the coin, how high it was flipped, etc.), we could predict the outcome with certainty. If we knew how the cards were arranged before the deal, we would know what our hand would be. But, unless we are cheating, we don’t know these things, so we have to rely upon probability assessments. In this case, probabilities are said to be epistemological.

But there are cases in which it could be argued that no amount of prior knowledge would allow us to predict the outcome – that probability is a genuine feature of the physical world. This is metaphysical probability, and it primarily comes up in the context of quantum mechanics and predicting the actions of Olivia, my sweet cat. V

Steve Humphrey has a Ph.D in the history and philosophy of science, with a specialty in philosophy of physics. He teaches courses in these subjects at the University of California, Santa Barbara, and has taught them at the University of Louisville.