The Birthday Paradox
I read something the other day that piqued my curiosity:
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 or 90%
The probability of 3 people choosing different marbles, or having no collisions, is:
10/10 x 9/10 x 8/10 = 0.72 or 72%
This same logic can be applied to having the same birthday. The probability that two people do not have the same birthday is:
365/365 x 364/365 = 0.99726 or 99.73%
Subtract that from 1 and you get the probability that two people have the same birthday:
1 - 0.99728 = 0.00273 or 0.27%
To make this exercise simpler, I wrote a quick python script for the calculations:
from __future__ import division 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 )
The output was surprising:
01 people - 00.00 percent 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
23 people in a room, 50.73% chance for a same birthday. It's really true.
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 almost1 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
1: The precision is beyond the computer's capability at this point so 99.99999.. is rounded to 100. To truly guarantee a same birthday, there must be 366 people (or 367 on a leap year).