Explicit formulas in toy models

The human animal differs from the lesser primates in his passion for lists of "Ten Best". H. Allen Smith

This page outlines part of a program to articulate what mathematical probability says about the real world, aimed at students who have taken one course in mathematical probability but who have not taken a course on stochastic processes. The purpose of this part is to give a quick overview of what applied modeling claims to do, by presenting examples of the following kind.

(i) A mathematical model that's easy to describe.
(ii) An explicit formula for some aspect of the model;
(iii) an aspect that a non-mathematician might care about.

In particular, in this part of the program we do not

(a) analyze the realism of the model
(b) attempt a mathematical derivation of the formula.

Think of all this as a quick tour of some highlights of classical applied probability. Listed below are the examples I have used in my undergraduate course. I don't have any great emotional attachment to these particular examples -- I am just seeking to sample from different topics in the broad range of applied probability, without attempting any kind of completeness. Suggestions for further formulas are welcome.

The list

Envisaging a reader familiar with classical applied probability, I have just written brief phrases.