Birthday paradox

[Obrochta]

So I have always heard that out of a room of 23 there is a 50% chance that 2 people share a birthday. Out of 70 there is a 99.9 chance. Tonight I was in a bar of 50 and I shared a birthday with someone. Which was expected. But we were the same year as well. How rare is that with those same odds of 23 random people and 70 random ppl.

 

[Deleted member 4993]

So I have always heard that out of a room of 23 there is a 50% chance that 2 people share a birthday. Out of 70 there is a 99.9 chance. Tonight I was in a bar of 50 and I shared a birthday with someone. Which was expected. But we were the same year as well. How rare is that with those same odds of 23 random people and 70 random ppl.

Bar-crowd age (birthday) is not totally random. Lower limit is obviously 0 - could be 18. Sensible upper limit could be 100.

 

[Cubist]

Using the age range suggested by @Subhotosh Khan (82 years), I calculate the following probabilities...

Number of people	Probability of shared birth date (year and day)
50	0.040098
204	0.499888
642	0.999012

Spoiler: Show python code

import math
n=205
slots=math.floor(365.25 * 82)
p=1
for i in range(1,n):
    p *= (slots-i)/(slots)
print(1-p)

 

[Steven G]

If everyone in he bar is say 21, then if two people share the same birthday then they will share the same year. The bar room crowd as SK pointed out is not random. I suspect that the age of the crowd will follow a bell curve rather than a uniform curve. This is why your question can not easily be answered.

 

[Cubist]

...I calculate the following probabilities...

...I assumed a uniform distribution across the 82 years with this calculation. @Jomo and @Subhotosh Khan are correct, some bars will have die-hard regular drinkers ? whose birth dates won't change day to day. And many bars will attract more people from a certain age range depending on the music (or lack thereof), decoration, location, etc. So don't take my results too seriously!

It's probably worth pointing out that in post#1, "out of a room of 23 there is a 50% chance that 2 people share a birthday", also assumes a uniform distribution of birth dates. However, type "birth date distribution" into a search engine. Mid September is more common than other dates in America and UK. Taking such distributions into account will make it more likely that a match will occur.