Algebra for Adulterers
What every philandering politician and traveling salesman should know about the odds of getting caught.
The fact of the matter is that intimacies tend to come in clusters. If Mom 1 is close to Mom 2, and Mom 2 is close to Mom 3, it becomes more likely that Mom 1 and Mom 3 share secrets over the fence as well. Moreover, one might predict the presence of "intimacy hubs"—a small number of women who talk freely to a very large number of others. This kind of clustering behavior is quite common in real-life networks like airline flight maps and the World Wide Web. Researchers such as Albert-László Barabási of Notre Dame have developed quite sophisticated models for networks with these clustering properties. We'd like to use the work of Barabási and his colleagues here but two problems intervene: First of all, there's no empirical evidence that the networks of intimate acquaintance are of the kind Barabási studies. Second, even if the intimacy network were of this kind (and it seems reasonable to guess that it is), we don't know how, without the product rule, to estimate the probability that no two mothers in our sample are intimate.
To sum up: The independence assumption is false, and we shouldn't apply the product rule.
Next step: Go ahead and apply the product rule. Remember, there are about (1/2)g2 pairs of moms, and each one has a (1 - n/N) probability of being non-intimate. So, the chance that all pairs of moms are non-intimate is about
According to the Census Bureau, congressional districts contain an average of 28,000 18- to 24-year-old women each. Let's say their mothers have an average of five intimate friends apiece. Then the chance of the congressman avoiding any unpleasant revelations is
If the congressman has 10 young girlfriends, this comes out to a 99 percent chance. If he has 20, a 96 percent chance. If he has 50, an 80 percent chance. The loving legislator would have to romance 88 of his constituents before he'd create a 50-50 chance that two moms would talk.
Now that's just the kind of spurious precision I was talking about. The numbers above depend not only on the BFA but on our smaller false assumptions about mothers and their conversation habits and on our outright guess about the average number of intimates of each mother. So, what can we really say confidently? One is probably on safe ground saying that if the congressman has a dozen or two dozen girlfriends, there is a small chance but not a trivial one that he'll be caught out. Not one in a million but also not one in two. That's a wide range, but it's what honesty demands.
We can say somewhat less under our breath that if the congressman wants to keep his chance of discovery down to 5 percent, the number of girlfriends he can afford should vary in proportion to the square root of the number of eligible young women in his district. For instance, a promotion from the House of Representatives to the U.S. Senate would increase the district population by a factor of 57, which means our protagonist could keep a black book seven times as thick without increasing the risk the moms would meet. (By contrast, a move from the Texas governor's mansion to the White House allows for less than a fourfold increase in extramarital affairs.)
There's one more assumption we haven't mentioned: that congressmen mate randomly. Do they? The good thing about ultra-rough computations like ours is that we can safely say: close enough.
Jordan Ellenberg is a professor of mathematics at the University of Wisconsin. His book How Not To Be Wrong is forthcoming. He blogs at Quomodocumque.