# Happy Birthday To Me!

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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