In a random gathering of 23 people, there is a 50% chance that two people will have the same birthday.No way, I've been to many gatherings and I don't remember anyone having the same birthday. Then again, I never really checked..

Thus, began my journey to verify this madness. I messed around with permutations and combinations for half an hour, only to leave more confused than when I started. A new approach was needed.

I decided to look at the problem from a different angle. I could calculate (more easily) the probability of

**no collisions**, then subtract that from 1 to get the number of

**collisions**!

Say there are 10 marbles in front of you and each person has to pick a different marble. The first person has 10 marbles to choose from. The second person only has 9 to choose from. So the probability of 2 people choosing different marbles is:

10/10 x 9/10 = 0.9 orThe probability of 3 people choosing different marbles, or having90%

**no collisions**, is:

10/10 x 9/10 x 8/10 = 0.72 orThis same logic can be applied to having the same birthday. The probability that two people do not have the same birthday is:72%

365/365 x 364/365 = 0.99726 orSubtract that from 1 and you get the probability that two people have the99.73%

__same__

**birthday:**

1 - 0.99728 = 0.00273 orTo make this exercise simpler, I wrote a quick python script for the calculations:0.27%

from __future__ import divisionThe output was surprising:

import sys, math

def CalcProbMatch( n, days ):

prob_no_match = 1

for i in range( n + 1 ):

prob_no_match *= (days - i)/days

prob_match = 1 - prob_no_match

print '%02d people - %05.2f percent' % ( i, prob_match * 100 )

for i in range( 27 ):

CalcProbMatch( i, 365 )

01 people - 00.00 percent23 people in a room, 50.73% chance for a same birthday. It's really true.

02 people - 00.27 percent

03 people - 00.82 percent

04 people - 01.64 percent

05 people - 02.71 percent

06 people - 04.05 percent

07 people - 05.62 percent

08 people - 07.43 percent

09 people - 09.46 percent

10 people - 11.69 percent

11 people - 14.11 percent

12 people - 16.70 percent

13 people - 19.44 percent

14 people - 22.31 percent

15 people - 25.29 percent

16 people - 28.36 percent

17 people - 31.50 percent

18 people - 34.69 percent

19 people - 37.91 percent

20 people - 41.14 percent

21 people - 44.37 percent

22 people - 47.57 percent

23 people - 50.73 percent <-

24 people - 53.83 percent

25 people - 56.87 percent

26 people - 59.82 percent

27 people - 62.69 percent

In hindsight, it's similar to the penny/wheat and chessboard problem in that we don't see how quickly the compounds or "combinations" can grow. When I first saw this problem, I pictured the probability that 23 other people would have the same birthday as me (pretty low actually). What I failed to consider, however, was that the statement also includes comparing other people's birthdays with each other.

Anyways, it was a fun experiment and I learned something new.

Bonus: How many people would have to be in a room to guarantee a same birthday?

80 people - 99.99 percent

81 people - 99.99 percent

82 people - 99.99 percent

83 people - 100.00 percent <-

84 people - 100.00 percent