Confusions: Probability

A current posting at Mathematical Association of America is about probability theory and how we often misunderstand it.

Keith Devlin makes the point that people have problems when they want to assign a probability to knowledge:

"In my experience, it's when probabilities are attached to information that most people run into problems.

The concept of probability you get from looking at coin tossing, dice rolling, and so forth is generally referred to as "frequentist probability". It applies when there is an action, having a fixed number of possible outcomes, that can be repeated indefinitely. It is an empirical notion, that you can check by carrying out experiments.

The numerical measure people assign to their knowledge of some event is often referred to as "subjective probability". It quantifies your knowledge of the event, not the event itself. Different people can assign different probabilities to their individual knowledge of the same event. The probability you assign to an event depends on your prior knowledge of the event, and can change when you acquire new information about it."

Devlin gives a beautiful example of subjective probability. I have my own though. Suppose you have ten beans covered by cups. Nine beans are black, one is red. What is the probability that when you turn over a cup the bean is red? Intuitively it's 1/10. Now, suppose I turn over nine cups so you see the covered beans. What's the probability that the last cup has a red bean? At this point it's no longer a probability but rather a statement of known fact. You can see the other beans. If one of those is red, the cup has a black bean. If none of the nine exposed beans is red, the covered bean is red. There's no uncertainty at all.

I've often wondered just what probability meant and how we can use it consistently and properly. Devlin makes a good start to doing just that.