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Birthday pairings

January 2, 2013

Re my three problems about birthdays from a couple of weeks ago…

Each solution needs quite a different way of looking at the problem – very common when dealing with different ways of arranging people or objects (an area of maths called permutations and combinations). This can make such problems fascinating and frustrating at the same time.

(Each problem also deserves its own discussion, so I’m giving each answer its own post…)

So, how many people must be in a group before there is a better than 50% chance of having at least two people in that group with the same birthday?

In the discussion below, I ignore both of the following in the general analysis, but comment on their influence on the problem outcome.
– February 29
– Some example data here shows that frequencies of birth dates are spread over a range of about 20% from highest to lowest, excluding a handful of extra low frequency dates (e.g. Jan 1, Jul 4, Dec 24, Dec 25). There also seems to be a weekly cycle, with about 2% less on weekends – presumably when doctors are playing golf.

With many problems like this, where the question has the form “At least one …”, it’s often easier to first work out the probability of the opposite problem. Here this means no pairings – no people with the same birthdays.

So, to have no pairings…
– the first person can choose any date they like
– the second can choose 364 out of 365 dates
– the third can choose 363 out of 365 dates
– the fourth can choose 362 out of 365 dates
etc

So the probability of no pairings is
365   364   363   362   …
— x — x — x — x —
365   365   365   365   …

When does this fraction dip below 0.5 = 50%?

Perhaps rather unexpectedly, it takes only 23 terms until we have

365   364   363   362   …   344   343
— x — x — x — x — x — x — = 0.493
365   365   365   365   …   365   365

That is, for 23 people, there is a 49.3% chance that there will be no pairings, and so a 50.7% chance that at least 2 people will have the same birthday.

The probabilities for at least one pairing is 71% for 30 people, 89% for 40 people and 97% for 50 people.

To the extra factors mentioned above:
– Taking February 29 into account increases the average number of people required from 22.8 to 23.0 people. So we still need only 23 people.
– The link here – the same as that above – shows that the different frequencies don’t affect the probabilities much (a little surprisingly to me, given the range of frequencies).

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