The point that human beings are terrible at understanding probabilities is often made by psychologists and economists. This talk by Dan Gilbert, at TED 2005 Global, is a good example. ‘Why would anybody play the lottery?’ he asks, adding that economists refer to the lottery as a stupidity tax. (A member of the audience points out in the Q&A session that people play the lottery not just for the money, but for the thrill of playing, which economists unseemly disregard, but never mind that).
We are pathological comparison-makers, Gilbert says. Compare George Bush to Junior, and suddenly he doesn’t look so bad! We invariably think not of how much money we’re saving, but how large a fraction of the cost we’re saving. Koutons marks its jeans up by 400% and gives you an 80% discount (some quick arithmetic should tell you that there’s no net change in the price), and that seems to us like a better deal than some other manufacturers who do nothing to the marked price. So, it seems that in spite of being comparison-maniacs, we suck at making the right comparisons, and at getting our comparisons right. There are many more of these examples in the talk, which is informative and thoroughly entertaining.
In lieu of changing how you think about things or make comparisons, unlike Dan Gilbert, I’ll settle for showing you a calculation of probabilities!
How many friends do you know who share birthdays? To put that in more understandable terms, perhaps, how many times have you looked at your Facebook page and seen more than name in the ‘Today’s Birthdays’ column? A few? Good. How improbable do you think it is, assuming that when somebody is born is approximately random (I’ll come back to this at the end), that people you know share a birthday? In other words, how many people do you have to know before the chance that there will be a common birthday will be at least as large as the chance that there won’t?
Would it surprise you to learn that the answer is 23? Here’s a graph of the probability that there will be at least one shared birthday in your group of friends versus how large your group is:[prob]
At 23 people, the probability is better than even (equal to 0.507) that there will be at least one shared birthday. So unless you’re a complete loner, it shouldn’t be surprising that you have friends who share birthdays.
Now, the uniform probability assumption isn’t strictly true. Birthdays tend to be clustered. (Go figure.) I’m not sure how much of a difference this makes, but it doesn’t seem to be much.
Also, this is only the probability that there will be at least one shared birthday. This does not tell you how many shared birthdays you should expect given that you know N people. That is a much harder problem which I haven’t quite been able to write down an answer for.