Misunderstandings arising from different perspectives on probability

There are four perspectives on probability that are commonly used: Classical, Empirical (or Frequentist), Subjective, and Axiomatic. Using one perspective when another is intended can lead to misunderstandings and errors.

Various authors¹have pointed out that students' first formal introduction to probability, whether in middle grades, high school, or an introductory college statistics course, is often from the classical perspective, where we talk about outcomes (such as the numbers that can come up when tossing a fair die) that have equal probabilities. However, many applications of probability (including statistical inference) involve situations where outcomes may not have equal probabilities. This can lead to misunderstandings such as the following:

Common misunderstanding: If there are only two possible outcomes, and you don't know which is true, the probability of each of these outcomes is 1/2.

In fact, probabilities in such "binary outcome" situations could be anything from 0 to 1. For example, if the outcomes of interest are "has cancer" and "does not have cancer," the probabilities of having cancer are (in most cases) much less than 1/2. The empirical (frequentist) perspective allows us to estimate such probabilities.

Notes:

1. See Albert, James H. (2003), "College Students' Conceptions of Probability", The American Statistician, vol. 57 No. 1, pp. 37 - 45, and references therein.