Ed Barbeau, who is possibly a little exhausted from the doomsaying on this blog, has pointed out to us a recent BBC article on the birthday paradox. By mathematician Kit Yates, the article is framed around matching soccer players’ birthdays at the recent Women’s World Cup. There’s nothing grandly original about Yates’s article but it’s very nicely done, and it may form the basis of a good classroom discussion and/or assignment.

Regular doomsday programming will continue tomorrow.

The birthday paradox is certainly one of the more pleasant things to mention, if not to teach, to students in high school (where I must say I lack the teacher perspective) and early uni years. (BTW, I was pleased to note many years ago that somebody knew the answer to the paradox question in a uni class with little maths background. How widely it was actually known remained unclear to me.) The BBC article does offer decent arguments that it is more than a feel-good digression.

As a point of criticism, besides the typo in the caption that Marty gracefully cut out (yeah, I am pedantic), I feel the article ends on a somewhat weakish note. I have four items regarding it:

First, the statement, “… even if the chance of any one of them [i.e., what he calls the “possibilities for an event”] occurring seems low …” is confusing in my view. In the present instance, the possibilities that make the event “two players in a team share a birthday” happen are that Player 1 shares her birthday with anyone else in the team; Player 2 shares her birthday with anyone else in the team; and so on. The probabilities of the foregoing events _each by themselves_ do not only “seem” low, they “are” low. (Somewhere around the 1/365 mark but then some birthdays, such as mine, are more common for reasons I don’t currently know about, but that is a secondary matter. Also, the possibilities above are not disjoint or mutually exclusive events – say if Player 1 shares a birthday with Player 3, then Player 3 of course shares a birthday with Player 1 –, but that is again secondary.) So I’d just replace “seems low” by “is low”.

The second point is that, in the ideal world at least, the teacher should be aware that “by chance” is often, such as here, used to describe events that occur with equal probability. (Same goes for “at random”, “randomly”, etc.) The equal probability assumption is not always realistic – see the parenthetical remark in the previous paragraph – and in fact not necessary for the paradox to occur; this “robustness” of the paradox, so to speak, should not be out of reach for many students, once they have mastered its essence.

The third point is that it would have been nice to remind the reader of the less extreme situation where a seemingly highly improbable event, by virtue of the birthday paradox, becomes not overwhelmingly likely, but “only” much less improbable. One way to approach these issues is to recall the curve from the beginning and ponder what it says in its entirety (a suitable classroom task, maybe). For example, What is the situation if there are 18 students? Or just 7? (Take out your ruler and measure these probabilities from the graph, or perhaps if possible, compute.) And there are indeed cases where even “somewhat improbable” is enough to make a related decision that would have been very different if the event in question had originally been considered as highly improbable.

My fourth and final point is that understanding the birthday paradox does not obviate the need to look at stochastic or statistical dependencies in real-world cases. Soccer team players may have their birthdays thrown together in a completely “mixed” fashion, but other matching of pairs may, if subtly, be influenced by birthdays and this may be “on top” of the birthday paradox, or even the most important (and harder to resolve) part of the data situation.

Correction: „Player X shares her birthday with anyone else in the team“ has a probability around (size of team)/365 and not 1/365. My apologies.

One person I sent the article to mentioned that the problem was mentioned on one of the US late night shows hosted by I believe Johny Carson. He was with a studio audience of about fifty and he portrayed the situation as being almost certain that two people in the audience would share *his* birthday. As my correspondent said “lead balloon”. On one occasion with a class of about a couple of dozen, there were three who shared the same birthday.

This does raise the issue that often with such things, it is difficult for the general public to understand what the problem *is*. This is the case for example with the four colour problem.

With a class of students of about the same age. you could even consider how likely it is that two were born on the same day.

The article is definitely a nice application of an existing mathematical problem to the relevant context. Question to you Marty. Let’s say it wasn’t written but explained in a video. Will you still consider it as an example that “may form the basis of a good classroom discussion and/or assignment”?

No.

I once taught this problem to my students and then sent them through the whole school to ask in each class if there are two students sharing a birthday. Besides coming back with almost the expected number, they also came up with an interesting modification of the question.

Suppose that in a class of n students two share a birthday. What is the probability that they are twins?