Explicit formulas in toy models

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

• Mean duration of fair game (SRW): x(N-x)
• M/M/1 queue mean length: ρ/(1- ρ) [ supermarket checkout line]
• Effective number of neutral alleles: (1 + 4 N θ ) [ 3 hair colors, not 33]
• Just-supercritical GW survival probability: 2(μ - 1)/σ² [how infectious must an epidemic disease be?]
• Kelly diffusion: P(portfolio sometime drops below r times initial value) = r. [Was the 2008 market decline so unusual?]
• Black-Scholes: $C(S,t)\; =\; SN(d\_1)\; -\; Ke^\{-r(T\; -\; t)\}N(d\_2)$ where $d\_1\; =\; \backslash frac\{\backslash ln(S/K)\; +\; (r\; +\; \backslash sigma^2/2)(T\; -\; t)\}\{\backslash sigma\backslash sqrt\{T\; -\; t\}\}$ $d\_2\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\{T\; -\; t\}.$
• 7 shuffles suffice ! [from exact formula in Bayer-Diaconis]