Today's example comes from a World Science Festival discussion entitled "The Illusion of Certainty", a little symposium on risk, probability, and humans' frequent (sometimes innate) inability to properly assess said probabilities and risks. (Note: seriously, don't bother watching that whole video stream--it's a ridiculous nerdfest... ridiculous.)
At one point, one of the panelists stood up and addressed the room (there were roughly 50 spectators at that point in the show--not sure if they started with more and some tapered off, tired of the nerdfest, but I digress...), and told them that he could say with extreme certainty that at least two people in that room shared the same birthday.
The crowd of course doubted the panelist, as did I. With 365 (yeah, okay, 366 if you include February 29) possible birth dates and only 50 people in the room, it just didn't seem like something that was particularly likely. Then, he began randomly asking spectators their birthdays. Amazingly (and he couldn't possibly have planned it this way, this was just dumb luck), the first person he asked had the same birthday as him. He asked about 10 more people in the room for their birthdays, and already he had found another match. He didn't even have to go any further to show that the intuitions of all in that crowd were dead wrong.
As it turns out, this is one of those instances where the science and our intuition are simply in dead opposition with each other. Most of the other panelists (all of whom, mind you, were statisticians) even agreed that it would seem like the crowd would need to be at least 100 strong in order to have any certainty of a birthday match. But the math disagrees.
As it turns out, in a group of only 22 (or 23, depending on how you want to do the math), the probability of a birthday match has already reached 50%--in other words, even odds. By the time you reach 50 individuals in the crowd, the probability swells to an amazing 98%--hence the panelist's extreme confidence.
Even after doing the math myself, I really didn't even believe that it made sense. I needed further convincing, so I went and performed a little "experiment". If the math was correct, and any randomly-selected group of 23 people had about even odds of having at least two people with the same birthday, then I should be able to look at any list of 23 or more individuals and expect a match roughly half the time.
Where could I find such a list? Major League Baseball rosters, of course. 30 teams, each with 25 players--according to the math, with 25 players per team, the probability of a birthday match should be about 60%, which would mean 18 of the 30 teams should have at least one birthday match.
So what did I find?
For those of you that don't feel like counting, that's 16 out of 30 teams (53%) with at least one match. Three teams (A's, Diamondbacks, Indians) had multiple birthday matches, and two teams (Mets and Orioles) had birthday "triplets". The names in bold had the same exact birthdate (same year), which was even more impressive.
So there you have it. The intuition might not lead you there, but the data is definitely fairly close to what the math would predict. I'm not nearly patient enough to do this same exercise for football teams, but given their 53-player active rosters, the math suggests that over 98% of teams (in other words, nearly all of the 32 NFL teams) should have a birthday match. Anyone feel like checking into that one for me?
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