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.