Abstract
“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always of the same gender as the member making the selection. This model for evolution is isomorphic to the classic Pólya’s urn. The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, then replacing it and adding a new ball of the same color. It is well known that for Pólya’s urn, the proportion of red balls in the urn is a martingale. It follows that for a consanguinity, the proportion of the membership that is male is a martingale. Furthermore, being bounded, this martingale converges to a limit. For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is also a well-known second-degree martingale from which the variance of the limiting distribution can be deduced. What the author discovered, in the process of solving his own examination problem, is that a similar martingale exists also for Pólya’s urn, even though in this case the number of red balls is the sum of random variables that are not independent. This new martingale can be used to calculate the variance of the limiting distribution. Traditionally, the probability that r red balls will be drawn from Pólya’s urn in n trials is derived by a rather tricky argument involving conditional probability. This article uses an obvious but overlooked simpler approach. Pólya’s formula for the probability that m male members will be chosen in n weeks is derived, without any mention of conditional probability, by an elementary counting argument, and its limit is shown to be a beta distribution.
Highlights
“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members
The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, replacing it and adding a new ball of the same color
For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is a well-known second-degree martingale from which the variance of the limiting distribution can be deduced
Summary
There is a long tradition of setting unsolved problems on mathematics examinations. The most famous example is Stokes’s Theorem, a three-dimensional version of which appeared on the 1854 Smith’s Prize Exam at Cambridge University [1], several years before a proof was published by Hankel. Instead of a consanguinity with male and female members, they considered an urn which initially, for the special case in the exam problem, contained R = 1 red balls and S = 3 ( S for “Schwarz”) black balls. Since the students had done examples based on Pólya’s urn and had studied the martingale convergence theorem, everything was straightforward except for “describe the probability distribution for μ∞ ,” which the author of the exam question had not taken the trouble to work out!
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
