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Old 04 October 2012, 01:45 AM
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TV The Birthday Problem

A classic puzzle called the “birthday problem” asks: How many people would be enough to make the odds of a match at least 50-50?

The answer, just 23 people, comes as a shock to most of us the first time we hear it. Partly that’s because it’s so much less than 366. But it’s also because we tend to mistake the question for one about ourselves. My birthday.

John Allen Paulos gave a vivid example of this error in his trenchant best seller “Innumeracy”:

A couple of years ago, someone on the Johnny Carson show was trying to explain [why 23 is the answer to the birthday problem]. Johnny Carson didn’t believe it, noted that there were about 120 people in the studio audience, and asked how many of them shared his birthday of, say, March 19. No one did, and the guest, who wasn’t a mathematician, said something incomprehensible in his defense.
For years I’ve been dying to see a clip of that scene. It’s become legendary, an iconic moment in mathematical pop culture. Yet I couldn’t find it on YouTube, and none of my colleagues, including Paulos, could remember when it occurred or who the humiliated guest was. Memories have understandably faded over the years.

But fortunately, as of 2010 every surviving tape of “The Tonight Show With Johnny Carson” has been digitized and made publicly available. Thanks to the Carson archives and some ace detective work by a researcher at The Times, we can now watch what really happened when Johnny met the birthday problem. The encounter offers lessons not just about math but about memory as well.
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