Probability models

Probability distributions can serve this purpose quite well. There are many fundamental types of distributions, but the Poisson distribution is a good one to discuss first because it is appropriate in situations where there are discrete occurrences of things with respect to time:

There are three different variants of the distribution shown here, each with a different mean (λ), and the highest expected value of k. We can make an analogy that says that these distributions model the expected amount of postal mail that a person gets delivered to their home on a daily basis, represented by k on the x axis:

• For λ = 1, there is about a 37% chance that zero pieces or one piece of mail is delivered daily. Perhaps this is appropriate for a college student that doesn't receive much postal mail.
• For λ = 4, there is about a 20% chance that three or four pieces are received. Seemingly, this is a good model for a young professional.
• For λ = 10, there is about a 13% chance that 10 pieces are received per day—perhaps representing a larger family or at least a household that has somehow found themselves on many mailing lists!

The discrete points on each curve also give the likelihood (probability) of other values of k. As such, the model can be informative and answer questions such as "Is getting fifteen pieces of mail likely?". As we can see, it is not likely for the student (λ = 1) or the young professional (λ = 4), but it is somewhat likely for the large family (λ = 10).

Obviously, there was a simple declaration made here that the models shown were appropriate for the certain people described—but it should seem obvious that there needs to be a mechanism to learn that model for each individual situation, not just assert it. The process for learning it is intuitive.