The Birthday Paradox

Recently two of my friends (and neighbours) in the village noticed that they shared a birthday. They expressed surprise at this revelation because it seems improbable. Actually it is likely that a relatively small group of people will have two members with the same birthday. For instance, with groups of 25 people, two will have birthdays on the same day of the year in 56% of cases; with groups of 60 the likelihood increases to more than 99%. These apparently unlikely facts can be demonstrated by some simple maths.

The simplest way to start this is to investigate the likelihood that people do not have the same birthday. The probability t hat two people do not have the same birthday is 364 / 365 , as there is only one chance in 365 that one person's birthday will coincide with another's). The probability that a third person's birthday will differ from the other two is 363 / 365; a fourth person's 362 / 365; a fifth person's 361 / 365; ans so on to the fraction for the 25th person which is 341 / 365. This gives us 24 fractions that have to be multiplied together to reach the probability that all 25 birthdays are different. This gives a percentage of 43.13%. Given that there are only two alternatives, either no birthdays match or two (or more) do match. Therefore the probability that there will be a match is 100 - 43.13 = 56.87%

I calculated these percentages in a spreadsheet. The formulas that I used are shown in the picture below. (Naturally, just enter them in row 3 and fill down.)



For a larger and clearer version click on this link.
I also charted the spreadsheet. The probabilities can be read off the chart below:



For a larger and clearer version click on this link.

So what to make of my friends' birthday coincidence. There are eleven people living in our street. If we just consider this group the birthday coincidence is relatively unlikely at 14% though not at all impossible. If you consider the wider village with almost 150 people it is virtually a certainty that two people will have the same birthday. Although I haven't checked, I expect that there is likely to be at least one other birthday coincidence.