My last blog post illustrated how unlikely the equal birthday likelihood assumption is. I wrote a short simulation code to consider the impact of unequal birthdays. I modeled unequal birthdays as a mating season that results in three months (90 days) that are more likely birthdays than the remaining 365-90 days. This corresponds to July – September in my earlier post.
Let R = (Likelihood of being born in the “hot” 3 months) / (Likelihood of being born in the remaining 9 months).
The Birthday Problem assumes that R = 1. I consider 1 <= R <= 2. This post courtesy of Chris Rump indicates that R < 1.2, meaning that humans don’t have much of a mating season.
The simulations below show the average value of P(n), where P(n) = the probability that someone shares a birthday in a group of n people. The simulations are performed over 1M replications for each value of n. The probability of shared birthday goes up when people are more likely to be born in the birth months associated with “mating season.” But the effects are small, as can be seen by a fairly compressed y-scale. The simulations were performed in Matlab and the program is here.
The Birthday Problem probability for n=5.
The Birthday Problem probability for n=10.
The Birthday Problem probability for n=20.
The Birthday Problem probability for n=30.
The Birthday Problem probability for n=40.
The Birthday Problem probability for n=50.
Punk Rock Operations Research 
May 21st, 2012 at 5:54 pm
So it appears that the assumption of R = 1(equally likely days) leads to a probability estimate that is off (too small) by about 1/2 % (if R < 1.2). Thanks for doing this!
May 25th, 2012 at 11:53 am
“Let R = (Likelihood of being born in the “hot” 3 months) / (Likelihood of being born in the remaining 9 months).
The Birthday Problem assumes that R = 1.”
If each month were equally likely, wouldn’t R=3/9=1/3?
May 25th, 2012 at 12:15 pm
@suevanhattum, Good point. I didn’t articulate this week. I was looking at a per day probability of being born. R = 1 only when someone has a 1/365 chance of being born in any day in the hot 3 months or in the 9 non-hot months.