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# When phenomena coincide, it may not be coincidental

9 May 2011

Perhaps you’ve been at a gathering and were asked to write the month and day of your birth on a piece of paper and hand it in. A sorter arranged all the slips in calendar order and, lo and behold, two or more coincided. How much of a coincidence was it? Even with a moderately sized group, the probability of the phenomenon is higher than you might guess.

You can ascertain the precise prospect of at least two members of a given-size group a sharing a birthday using a hand calculator or computer spreadsheet. The simplest way is to first find the chance of no matches. Then take the “complement,” the difference between this value and 100 percent, to get the likelihood that two or more were born on the same month and day.

To see how this works, picture a hat containing 365 slips numbered 1 to 365. The numbers represent days of the year in sequence (ignoring leap year). Say three individuals draw slips in turn. After each draw, the number is posted and the slip put back into the hat.

When the second person draws, the chance the outcome will deviate from the first is 364 out of 365 – that is, 364/365. Assuming the second does differ, on the third draw, the chance the result won’t correspond to either of the others is 363 out of 365 – that is, 363/365. The probability of three unique dates is therefore (364/365) x (363/365). Do the arithmetic to obtain 0.9918 or 99.18 percent. The chance at least two agree is then 100 - 99.18, or 0.82 percent.
The approach is the same for any size group. The terms multiplied together to get the chance of all dates differing start with (364/365) and go to 365 minus one less than the number in the group. So, for 50 people, the fractions are (364/365), then (363/365), then (362/365), down to (316/365) – 316 being 365 minus 49. If you don’t want to do the arithmetic, just trust me. The probability come out to 3.96 percent of no duplicates and the other 97.04 percent of at least one match. For 75 people, the chance of at least one instance of a common birthday is 99.97 percent.

The technique can also be used to investigate what may seem like either an amazing bit of fortuity or some hanky-panky in a lottery in Germany, although you’ll need more ciphering power to check it out than most hand calculators can muster. In the game, six numbers are drawn from a set of 1 to 49. There are 13,983,816 – roughly 14 million – possible combinations.

On December 20 1986, the winner was 15-25-27-30-42-48. The upshot was identical on June 21 1995, by which point the game had run 3,016 times. Since the outlook for any designated combination of six numbers is close to one out of 14 million, the duplication in just over 3,000 tries attracted lots of attention – not all of a kindly character.

To discover the chance of no repeats in 3,016 draws, multiply 13,983,815/13,983,816 times 13,983,814/13,983,816 times 13,983,813/13,983,816 times all the probabilities in descending order to 13,980,801/13,983,816 (13,980,801 being 13,983,816 - 3,015). The answer is 72.24 percent. Subtract this from 100 to obtain 27.76 percent likelihood the duplication would occur. This probability is hardly overwhelming, but it exceeds one out of four – which isn’t bad.
These results are counterintuitive because most folks tend to garble the question. They think of the chance more than one person in a group will have a particular birthday or that a selected six-number set will hit in the lottery more than once, rather than that two or more people will share any birthday or that any number will repeat. Johnny Carson made this error during a monologue on the Tonight Show. He mentioned his birthday and asked if anybody in the studio audience had the same one. Say there were 74 people present, 75 including himself. He expected a hit because his writers presumably told him the chance would be 99.97 percent, not a sure thing but close.

Imagine Carson’s chagrin in striking out. The trouble was that he didn’t ask whether any two or more of the 75 had a common birthday. He asked whether anybody else shared his particular birthday. Here, each member of the audience had the same 364/365 chance of a miss. So the likelihood none of the 74 would have this birthday wasn’t 364/365 times 363/365 and so forth down to 291/365, which would be a very low 0.03 percent. Rather, the probability was 364/365 multiplied by itself 74 times. This equals 81.63 percent, and the chance that at least one person would have the same birthday as Carson was 100 - 81.63 or 19.97 percent. Hardly negligible, but far from the 99.97 percent on which he based what he figured would be a memorable stunt.

Of course, you may ask the wrong question and get the answer to the right one. That would, indeed, be coincidental. When it happens in a casino, it can be a trap for solid citizens trying to apply probabilities to their gambling. As the Chaucer of chance, Sumner A Ingmark, observed:
Misinterpretation of a situation,
Causes some vexation, often great frustration.