How many people until two share a birthday?

Why the match comes so soon

Most people guess that a room needs well over a hundred people. They are thinking of someone sharing their birthday, and that does take 253 people for an even chance. But any pair will do. Twenty-three people make 253 pairs, and each pair is a fresh chance. With 23 people the chance of a shared birthday is just over one half. With 60 it is above 99%.

The mathematician J. E. Littlewood made the same point about miracles in 1953. "Improbabilities are apt to be overestimated," he wrote. Freeman Dyson later turned it into Littlewood's law of miracles. We notice about one thing a second for some eight hours a day. That is about 30,000 events a day, or a million a month. So a one-in-a-million event should happen to each of us about once a month.

Persi Diaconis and Frederick Mosteller looked at a front-page story about a woman who won the New Jersey lottery twice, called a "1 in 17 trillion" long shot. That was the chance for one person buying one ticket in each of two draws. The better question was whether anyone in the United States would win twice. Statisticians at Purdue found that better than even over seven years, and better than 1 in 30 over four months, the time between her two wins.

The same game in a trading room

Bill Miller's Legg Mason Value Trust beat the S&P 500 for 15 calendar years in a row, from 1991 to 2005. Michael Mauboussin, who worked with him, looked at how few funds beat the market in each of those years and put the chance at 1 in 2.3 million.

Leonard Mlodinow, in The Drunkard's Walk, said that was the wrong question. It asks about one fund named in advance. Asked about any fund, the chance was about 3%. Asked about any fund in any 15-year stretch over the decades, it was about 75%. A streak like that was more likely to happen than not.

Miller said it himself: "As for the so-called streak, that's an accident of the calendar." He called it "maybe 95% luck." Our floor has no skill at all, and a streak of 15 years turns up anyway in about seven runs out of ten. A record is only impressive once you know how many people were trying to set it.

Sources

1. Persi Diaconis and Frederick Mosteller, "Methods for Studying Coincidences", Journal of the American Statistical Association 84, no. 408 (1989), 853–861. The birthday problem and the double lottery winner.
2. J. E. Littlewood, A Mathematician's Miscellany (1953), chapter "Large Numbers". Freeman Dyson, review of Debunked!, New York Review of Books (2004).
3. Michael J. Mauboussin, More Than You Know, updated edition (Columbia University Press, 2007). Leonard Mlodinow, The Drunkard's Walk (2008). Ian McDonald, "Bill Miller Dishes On His Streak and His Strategy", Wall Street Journal, 6 January 2005.
4. The room and the floor are made fresh in your browser. Birthdays are drawn from 365 equal days. Each of the 3,000 managers beats the market in each of 40 years with a fair coin. The curve and the seven-in-ten figure are exact sums, not simulations.