Common Birth Dates asked for the expected number of pairs of common birth dates for a cohort of 1000 people whose ages were clustered around a certain value.
Now consider a similar question to the traditional birthday problem: If you attend a concert or other event where people tend to be of a similar age, how large would such an event need to be in terms of number of attendees, so that you'd have at least a 50% probability that there would be at least two people born on the exact same day (year, month and day)?
Again to make things specific: assume the standard deviation about whatever the mean age to be is 12 years and it follows a normal distribution.
Again feel free to vary the assumptions about the distribution.
I think my analytic expression is incorrect. Take the simpler problem of a "day of the year" birthday problem with a uniform distribution P(i) = 1/365. The expression for n people taking into account conditional probabilities is:
P(match) = 1-P(no matches in n people)
= 1 - (364/365)(363/365)...(365-n+1)/365) = 1 - n! (365 n)/365^n
which is 0.507 for n=23
My errant formulation (which ignores conditional probability) would be here:
ppair (one pair with no match) = Sum(m=1,365) (1/365)(364/365)
= 364/365, and so
P(match) = 1-(ppair)^(n(n-1)/2) = 1 - (354/365)^(n(n-1)/2)
For n=23 this is:
P(match) = 1 - 0.49952 = 0.50048
which is an underestimate this time..... a different answer nonetheless.
So for the analytic I need to consider an expression that uses conditional probabilities.....
Edited on November 19, 2021, 10:46 am
explanation of discrepancy
