What is the probability that 3 people in a room of 15 people have the same birthday?

Answer 1

Hi Carl B.,

There's a fairly standard approach followed on these kinds of problems. Rather than trying to "forwards" calculate the various combinations of birthdays which would coincide (which is quite involved, there's lots of permutations by which that COULD occur!), instead back into the problem by asking, what is the probability that 3 people DON'T.

In fact, the best route goes through the following calcs:

P(no 2 people have the same birthday, within the group) --->

P(if two or more people do have the same birthday, what is the probability that no 3 people (or more) do?

So, for the first calc:

The first person has some birthday (leave off leap years, you can figure that on your own). The second person has P(not the same) of 364/365. Successive persons have 363/365, 362/365, etc. since the birthdays are still random across all 365 days, but the pool of acceptable dates is diminishing!

So for the first calc it's P( 2 the same) = 1 - ((364/365)*(363/365)*(362/365)*....(350/365)) = 1 - 0.71639599474 ~ 0.2836 . (By the way, this rises to ~0.5 for a group of 23 people).

OK, now, given that that's the P of at least 2 coincidences of birthday, how to do the second calc:

you have a set of people now with at least 2 birthdays coinciding. Pick a, or the, pair that coincide, and reexamine your remaining crew. Again, you are looking to avoid getting 3 of a kind. Now, that could happen in 2 ways: you could get a third person with exactly the same birthday, sequential-step P = 1 - (364/365)^13 ~ 1 - 0.965 ~ 0.035, and overall P = 0.2836*0.035 ~ 0.00994 (about 1%); or among the remaining 13 people, there could be another cluster of at least 2 identical, etc. (i.e. a regression problem). But since these are independent clusters, <~1%*1% (you now have a slightly smaller group of 13, so the likelihood of the 2nd cluster is a bit smaller, you could calculate it if you wanted) is a negligible additional sequential event probability contribution, and the approximate answer stays at 1.0%.

--Cheers, -- Mr. d.