What are the chances of two people in a group of 30 sharing a birthday? You may not think that the chances are good but look at it mathematically. The probability of an event happening and the probability of the same event NOT happening always adds up to 1. So let’s look at the chance that nobody in the group shares a birthday with anyone else in the group. This turns out to be much simpler.

The second person in the group must have a birthday which is different from yours. The chance of this is 364/365. Then, the third person must have a birthday which is different from both yours and the second person’s. The chance of that is 363/365, since there are 363 days out of 365 which are not either your birthday or the second person’s, and so on.

To find the probability that all these are true, we multiply all these probabilities together. That’s 364/365 x 363/365 x 362/365 … x 336/365 which is the probability that the 30th person doesn’t share a birthday with any of the first 29, assuming those 29 all have different birthdays.

Working this all out we get .293684, which is the chance that NO two people in the group of 30 share a birthday. The chance that two do share a birthday is one minus this, i.e. .706316. That’s a better than 7 in 10 chance. Not a certainty, but more likely than most people would guess.

That sums it up.

## Thursday, 19 March 2009

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