# 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.
• 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: $C\left(S,t\right) = SN\left(d_1\right) - Ke^\left\{-r\left(T - t\right)\right\}N\left(d_2\right)$ where $d_1 = \frac\left\{\ln\left(S/K\right) + \left(r + \sigma^2/2\right)\left(T - t\right)\right\}\left\{\sigma\sqrt\left\{T - t\right\}\right\}$ $d_2 = d_1 - \sigma\sqrt\left\{T - t\right\}.$
• 7 shuffles suffice ! [from exact formula in Bayer-Diaconis]