Saxophone quartets and probability
Saxophone quartets consist of four saxophones, usually a baritone, tenor, alto, and soprano, or maybe a second alto instead of a soprano. Because all saxophones are essentially the same instrument, just at different sizes, the instruments blend remarkably well. Saxophone quartets are not common, and yet I've run into two in two days.
I arrived in Heidelberg on Saturday to have some time to walk around and get over my jet lag before the Heidelberg Laureate Forum (HLF) events started. While I was walking around I ran into the all-female sax quartet Femdüsax performing in an outdoor plaza.
Then the next day, there was another saxophone quartet performing at the opening ceremony for HLF. At first I assumed it was the same group I had seen the previous day. But this was a different group, Balanced Action. I spoke to a member of Balanced Action, and she was not aware of the other quartet Femdüsax.
What's the probability of seeing two all-female saxophone quartets in the same area on two consecutive days? It's a common question. Well, maybe it's not common to ask about saxophone quartets per se, but it is common to ask about probabilities following unusual events. And yet such questions are often difficult if not meaningless.
How would you approach the problem of assigning a probability to an event like this? You could simply say the probability 1 because it did happen. This approach argues that probability doesn't apply to events that have already happened, or at least it does not give interesting answers to such questions. But you could get around this obstacle by asking for the probability that such an event would happen again in the future. Under the frequentist interpretation of probability, you would imagine an infinite collection of similar universes and ask in what proportion of these universes would someone see two female saxophone quartets in two days.
Even if you're not bothered by imagining infinitely many universes, you then face more difficulties, stretching credulity a bit further. For example, what kind of person are we imagining running into the quartets? I have played in a saxophone quartet, and so my ears are tuned to pick them up. On Saturday, I heard the quartet from a distance and went to investigate. Someone else might not have bothered. And what do we make of the fact that both quartets are entirely female? Do we imagine that an equal number of men and women auditioned for both groups and in both groups only women won? Or do we imagine that in both cases a woman had the idea of forming a quartet comprised entirely of other women? And when we say someone runs into a quartet over two consecutive says, do we mean over a one-week time period, such as my visit to Heidelberg, or any two days over the course of a lifetime? These are just a few of many questions to answer.
If you take a Bayesian rather than a frequentist approach, you still have difficulties. Under a Bayesian paradigm, you could say that the data simply happened and are not random. However, there is a probability model lurking somewhere in the background, and the parameters of that model are random because they are not known with certainty. But what model is that? The former problems come back in a different form but still remain.
E. T. Jaynes once said that probability does not describe the world but only our understanding of the world. A probability statement must be embedded in a model to make sense. Sometimes that model is obvious and generally shared, and so it becomes invisible. For example, you can ask the probability of six coins all coming up heads without further explanation because it is commonly assumed that the coins are independent and that each has probability 1/2 of coming up heads. This may be a reasonable assumption, but it is an assumption. As situations become more complex or unfamiliar, it becomes increasingly difficult to construct a model, much less to gain consensus around one. Sometimes the task is so difficult as to be meaningless.
There is a continuum of difficulty between questions about coin tosses and questions about saxophone quartets. Probability applies most easily and productively toward the former end of the continuum, but is often requested to perform more toward the saxophone quartet end.