P(x|y) = eww, gross

I finished my first midterm of the semester this week.  There was a decent section that revolved around calculating P(x|y)....in other words the probability of x if we know that y has already happened.  In simple terms, if you have a probability of something happening (x), and something else that's related to x happens (y), the probability of x happening changes.

This is not an overly complex concept when it's spelled out mathematically, but in real life it can be hard for people to remember that improbable events are often far more probable if you consider what's already happened.

I was reminded of this when I was reading Dear Prudence (Slate.com's advice column) last week and came across this letter from a man (born by artificial insemination from an unknown sperm donor) who decided to seek out his father, only to discover that it was the same man who had donated to his wife's mother.  Oops.  And gross.  Seriously, gross.

What struck me as more though, was a response that was printed from another reader:

I know you/we cannot know, but color me skeptical that this letter is legit. The odds of such a “match” have to be very small. I can't help but wonder if this letter is a fiction pushing a political agenda.

This seems true of course...I mean, it's a sort of Casablanca moment right?  Of all the gin joints in all the towns in all the world....

But let's think about this couple for a second:

1. They'd be the same age.  I don't know about sperm banks in particular, but I do know that most stored bodily fluids are only considered "good" for a few years...so if two people were to result from one donor, they would likely be around the same age.  Most people tend to marry someone within a few years plus or minus their own age.
2. They'd be from the same place. Sperm donors are unlikely to trit trot around the country donating to various centers....they're more likely to stay in the city they actually live in.  I have no particular experience with this, but I'd imagine that people looking for a sperm bank stick to their same town as well....which means these two children would like be raised in the same city, making their chances of meeting go up and the culture they were raised in more similar.
3. Their mothers had a lot in common.  From the letter, the couple is at least over the age of 30.  From what I gather, it was less common for people to use sperm banks prior to 1980.  According to this article, in 1987, only 5000 single women asked for donor sperm.  At the time the letter writers mother and mother-in-law were looking, it was likely even fewer.  This means the two shared a very unique background, and had mothers who were both counter-cultural enough to go forward with this.  This would give them quite a bit in common.
4. They'd likely have at least some similar hobbies, interests, and personality traits. I think most people would agree that at least some of our hobbies/interests/personality traits/taste in friends/what have you are more nature than nurture.  I would thus think it extremely likely that two people sharing the same father would have at least one major hobby or interest in common, making it much more likely that their social circles would cross and that they'd have something to talk about when it did.  The more you believe genetics influence who you eventually are, the better the chances they'd meet.
5. People (might) like people who look like them.  I actually had some trouble finding a good paper that proves this, but it's a theory.  Interestingly, the only scholarly article cited in support of this on the Wikipedia page actually turned out to say they found no evidence of this.  This is why Wikipedia =/= research.
6. Their parents would likely have supported the union.  In-law issues are tough, but I'd imagine if you were a lesbian mother in the 80s and found out your adult son was going to marry the daughter of another lesbian mother from the 80s, you'd be pretty psyched.  Also, they'd very likely have been in a similar socio-economic class, as both their moms were well off enough to pursue this avenue.

So we're not calculating the probability of two random people finding each other.  We're calculating the probability that two people in the same age group, from the same city, with the same unique background and similar interests, from similar cultures, with some similar personality traits, who looked similar, would meet, start dating and with (likely) large amounts of family support get married.  The chances would still likely be small, but not nearly as small.  When you throw in that they ended up at the same college, the chances actually get pretty high.

Many people seems to think that we have some sort of "incest flag" that would make us not attracted to relatives, but it actually is more likely related to the Westermark effect...or being in close proximity during childhood.  While it appears sperm banks are now actually trying to account for this, there could be some scary ramifications for some people.  Kinda brings new meaning to the phrase "what's your name, who's your daddy?" eh?