Against All Odds

David Pacchioli
June 01, 1995

My friend George gives me a headache.

One Friday evening last fall we were standing in the bleachers at Jeffrey Field, watching a soccer game. The night air was pleasantly sharp, and the ring of bright lights cut a lush green oval from the surrounding darkness. You could smell the grass, hear the thump of boot against ball. It was mid-way through the first half, and we had just happened to meet Keith Ord, a professor of statistics at Penn State, who was standing in the next row, also enjoying the game.

cartoon soccer ball

Then George had to pose one of his questions.

"Dr. Ord?" he said, out of the blue. "In a low-scoring game like soccer, where there is a relatively high probability of a fluke outcome, it would be better to play a series of games against the same team, right? You'd get a better idea of which is the better team that way, right?

"Okay, but my question is, would a series of soccer games give you a better sample than a single game of football, where the probability of scoring is higher?"

George asks stuff like that. Not to show off, or even with a healthy gambler's interest. He was just wondering.

To Ord's credit, he gave George a funny look. (It was a Friday night.) But aloud he answered that actually George had hit on something. As it turned out, he explained, George's question reflected a basic point of issue among statisticians, one that had divided their ranks for centuries. It all had to do with there being two schools of thought on this business of probability.

That was when I moved over to the other side of the bleachers.

Too late: For the rest of the game I battled flights of mathematical fancy. Instead of watching the field, I found myself recalling youthful hours spent in idle calculation: the dice-baseball game I invented to pass the time on long car trips; the daily scouring of the racing charts in the old Philadelphia Bulletin. My most powerful lesson in probability had been brutally simple: the penny-flipping experiment in ninth-grade chemistry. I flipped and flipped, dutifully recording heads and tails in my lab notebook to the piercing strains of a Tom Lehrer record. (Lehrer was my chemistry teacher's idol, but that's another story.) It was like communing with some sacred law: The longer I flipped, the closer I got to the truth.

I continued to brood about probabilities over the next several days. The more I did so, the more they seemed to pop up. ("What are the chances . . . ?" "He's playing the percentages." "Defying all odds, . . .") And the more they showed themselves, the more I wondered, just what is a probability anyway?

"The probable is what usually happens," said Aristotle. Succinct. Here's another one, from a book called News & Numbers: "A probability is a calculation of what may be expected, based on what has happened in the past under similar circumstances." So.

But what about the two schools? Still curious, I went and bothered Ord during his office hours. The two schools, he was happy to answer, were the frequentist (or objective) and the Bayesian (or subjective).

A frequentist, he explained, is what I had been in ninth-grade chemistry. A frequentist arrives at a probability by dogged repetition. The probability of heads coming up in a coin toss is the number of heads that do come up divided by the total number of tosses, given a sufficiently large number of tosses.

That law I had felt myself bumping into long ago was the "weak law of large numbers," the frequentist cornerstone which says that given enough tries we get to the true probability. It lies out there, somewhere, waiting for us to sweep away the variability that obscures it.

Bayesian theory describes a very different sort of world. Here there is no "truth." Probability is a measure of belief, and for a given situation each person has his or her own. Thus two weather forecasters looking at today's conditions can offer different probabilities of rain for tomorrow, and officemates are willing to bet on the outcome of a football game. Their probabilities may converge as more data rolls in, but that's because their beliefs will have changed. "An argument advanced by the more adventurous Bayesians," said Ord, "is that this is how we go through life—with a built-in calculator continuously updating our beliefs."

The two approaches persist for the good reason that neither is completely satisfactory. The frequentist approach depends on the notion that the same experiment can be repeated, under identical conditions, a very large number of times. In practice, true repeatability is a tricky thing to achieve. Conditions change. On the other hand, the subjective approach is, well, subjective. It doesn't seem right that there wouldn't be at least a degree of certainty—a pattern, say—to the outcomes of a million or so throws of the dice.

In the case of George's original question, however, it's the subjective Bayesian approach that seems clearly superior.

First, since a soccer game falls into the realm of non-repeatable events, there's a question as to whether objective probability even applies.

If, however, you could assume repeatability, a frequentist would have to factor the advantage of playing multiple games against the different probabilities of scoring in soccer and in football. This in itself sounds like a complicated problem. "There's probably a higher proportion of fluke scoring in soccer and a smaller number of actual scores," said Ord, going along. "You'd have to account for both effects."

A Bayesian, on the other hand, confronted with a pesky probabilities question on a lovely fall evening, could, it seems to me, simply smile and say:

"I dunno, George. What do you think?"

Last Updated June 01, 1995