Happy Birthday To Me!

by Kevin J. Lin

With one human there is a 0 percent chance that you'll have two humans with the same birthday.

With two humans the probability that they won't share a birthday is 364/365. The probability that they will share a birthday is therefore 1 - (364/365).

With three humans the probability that they won't share a birthday is the same as for two humans, times 363/365. So the probability that three humans will share a birthday is 1 - (364/365) * (363/365). Notice that with each additional person added, the probability that he or she shares a birthday with one of the previous persons goes up, because there are fewer "free" days remaining.

Following this progression, the probabilities are:

Number of Humans	Probability of two shared birthdays
1	0
2	0.00273972602739725
3	0.00820416588478134
4	0.0163559124665502
5	0.0271355736997935
6	0.0404624836491114
7	0.0562357030959754
8	0.074335292351669
9	0.0946238338891667
10	0.116948177711078
11	0.141141378321733
12	0.167024788838064
13	0.194410275232429
14	0.223102512004973
15	0.252901319763686
16	0.28360400525285
17	0.315007665296561
18	0.34691141787179
19	0.379118526031537
20	0.41143838358058
21	0.443688335165206
22	0.47569530766255
23	0.507297234323986

So 23 humans will have a better than average chance of sharing a birthday.

What about our misunderstood friends the Martians, who have a year of nearly 670 days? (As someone in the discussion forums pointed out, they may have leap days as well, but it turns out not to matter.)

Surprisingly, only eight more Martians will be needed.

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