# What a probability means

My daughters and I went to a CoinStar machine last night to convert a huge bowl of change into an Amazon gift card. The question came up of the probability of the total coming out an even dollar amount. My oldest said this had probability 1/100. My second oldest asked why we could talk about probabilities at all since the bowl just contained whatever it contained. Without knowing it, they represented the two major schools of probability interpretation, subjectivist and frequentist.

Subjectivists use probabilities to represent degrees of human belief or uncertainty, as well as frequencies. A subjectivist would argue that while content of the bowl is not a random variable, our knowledge of that content is.

Frequentists shy away from such psychological interpretations of probability. A frequentist might come to the same 1/100 probability estimate but would interpret the statement as follows. “If we were to randomly fill the bowl with coins many times and take it each time to the CoinStar machine, in about 1 out of 100 trips on average we would end up with a whole dollar amount.”

To look at another example, suppose an Oxford librarian discovered a manuscript and suspected it of being a previously unknown Shakespeare play. He might analyze word choice frequencies to come up with a probability that the play was indeed written by Shakespeare. He might conclude, for example, that there is an 80% chance that The Bard wrote the play. This would be a subjective probability since the 80% figure is a statement about the librarian’s confidence, not about the manuscript itself.

A strict frequentist would object that this is all nonsense: either Shakespeare wrote the manuscript or he didn’t, so there’s no probability involved. He might try to salvage the probability interpretation by speculating about what would happen if we were to discover and analyze an infinite number of similar manuscripts.

## 5 thoughts on “What a probability means”

1. And where do YOU fall, John?… :)

2. Good question. I guess I’d have to say I’m a subjectivist since that’s the broader category. Subjectivists are willing to use probabilities to model uncertain knowledge, but they’re also willing to use probabilities as frequencies along with the frequentists.

At first I thought the subjectivist position was ridiculous: why be subjective when you can be objective? But then I realized that it’s not always possible to be objective. In that case you can either be explicit about your subjectivity, or keep it implicit and pretend it doesn’t exist.

3. Tom Hampton

“He might analyze word choice frequencies to come up with a probability that the play was indeed written by Shakespeare. ”

The way I understand this, the result of the analysis would *not* assign a probability to the Bard’s authorship, but rather the probability that the association between the word frequencies found in the manuscript and those used by the Bard would be expected to occur by chance. The distinction is of no small matter, in my opinion, and at the root of many problems for scientists, who believe that a very small probability that the association between two things would be due to chance vindicates their chosen hypothesis, e.g., that Shakespeare probably wrote the play. While this line of thinking is important to rule out the role of chance, it does not by itself advance the positive assertion that Shakespeare wrote it by a micron. For example, we might take p < .0000001 as proof positive only to later find the manuscript was forged by a scholar in 1935. How silly we would look…

Respectfully,

Tom

4. Im a bit confused. Could you elaborate? Im fairly new to probability theory, but I though the “two schools of thought” were frequentist and Bayesian. You mentioned something of this sort in another article of yours. And yet now you mention the subjectivists interpretation. Are there three? How does Bayesian fit into the mix?

I am under the impression that Bayesians believe in measurable parameters, such as physics, or counting up the sides of a die, in real world physical truths that they derive probabilities from… taken in the here and now as constants of the system. And frequentists look to long run behavior, historical documentation, expectation found through a series of attempts, empirically documented. A Bayesian will measure the weight of a die to determine if its balanced or if it favors one outcome, a frequentist will decide from observation.

Im just very confused about the various definitions of probability.

What if I said that the coin star question, the probability was 1/100… because there are only 100 possible distinct fractions of a dollar that we could arrive at. We can arrive at 0 cents surplus with the same likelihood of 99 cents or of 56 cents.

I would have called myself a Bayesian because I can count the possibilities up, and assuming they are all equally likely. If I ran the test many times to check, then Id be a frequentist. And yet subjectivists operate on a degree of confidence? Isnt that arbitrary? I could say that the probability is 50%… because that is how “confident” I feel.

5. Can you also interpret probability from a gamblers perspective? Expected value, what betting ratio constitutes a fair game? Is this also a frequentists perspective?