The birthday paradox predicts that in a group of 23 people, there is a 51% chance that two or more share the same birthday. This calculation assumes that birthdays are uniformly and independently distributed among the 365 days of the year, ignoring leap years.
It turns out that birthdays are not quite uniform. Some days have more birthdays than others, as shown in this heat map from Facebook’s Andy Kriebel. Oddly enough, September is the most common birth month.
Does this make any difference in our calculations? Roy Murphy collected data on birthdays from 480,040 insurance policy applications. Assuming that these frequencies represented the actual distribution of birthdays in the entire population, I ran a simulation to estimate the probability that at least two people in a group of 23 share a birthday. After 100 million trials, the number of successes was 50,784,080, which is in agreement with the theoretical probability. (My Python code is available here.)