Statistics Blog > Same Birthday Odds
It stands to reason that same birthday odds for one person meeting another are 1/365 (365 days in the year and your birthday is on one of them).
But consider this: If you get a group of 30 people together, two of them will almost definitely have the same birthday. This blew my mind when I was a student.
There were 30 students in my undergrad statistics class and the professor said the odds of two of us having the same birthday were very high. In fact, two people in the class did have the same birthday. This didn’t seem to make sense to me, as there are 365 days in a year.
My Initial (Incorrect) Reasoning
The odds are 1/365 that I will meet another person with the same birthday. But we’re not talking about just me in a class. We’re talking about every student having those odds. It’s like if I had a 1/10 chance of winning the lottery and I meet another person who also has a 1/10 chance of winning the lottery, then combined we have a 2/10 chance of winning the lottery. The odds of a “coincidence” increases with each person:
Me meeting a person with the same birthday: 1/365
Me and one other friend meeting someone with the same birthday: 1/(365/2) = 183
Three of us meeting someone with the same birthday: 1/(365/3) = 1/122
…
Twenty nine of us meeting someone with the same birthday: 1/12.
Those are pretty good odds, but not high enough to account for all those coincidences. That left me with a peculiar puzzle. The odds are actually much higher (over 100 percent for a class of 30).
The reason takes into account all of the possible combinations.
Why the Odds are Actually Much Higher!
One person has a 1/365 chance of meeting someone with the same birthday.
Two people have a 1/183 chance of meeting someone with the same birthday. But! Those two people might also have the same birthday, right, so you have to add odds of 1/365 for that. The odds become 1/365 + 1/182.5 = 0.008, or .8 percent.
Four people (lets call them ABCD) have a 1/91 chance, but there are 6 possible combinations (AB AC AD BD BC CD) so the probability becomes 1/91 + 6/365…and so on.
You can see how it isn’t quite as easy as just x/365!
An Easier Way to Calculate Same Birthday Odds!
If there are 30 students in a class, there are 435 ways two students can be paired. The odds of a “match” become 1/12 + 435/365…which is much greater than 100 percent.
Seeing as the odds are 1/365 that any two students will match birthdays and there are 3 possible matches, it’s no surprise that two of those students share the same birthday.
(Use the combinations calculator to figure the combinations out. It will also list all the possible name combinations if you really want to!).
Should I Perform This Experiment in Class?
I’ll be the first to admit I haven’t used this in class for the main reason that with 25 students in a class, the odds are a bit over 50/50 that this experiment will work. A second reason is that the above math is over simplified to be somewhat understandable. Even third or fourth year math majors will struggle a bit with the “true” probabilities behind why this works. Figuring out same birthday odds is very complex for many reasons including:
- More people are born weekdays than weekends; mostly due to C-sections and induced births happening during the week, when doctors prefer to work.
- Seasonal trends mean that more people are born in the summer than the winter.
Figuring out the true probabilities involves Bayesian logic; hop over to this Stanford University page for a more detailed explanation on Bayesian logic and same birthday odds.
References
McCown, J. & Sequeira, M. (1994). Patterns in Mathematics: Problem Solving from Counting to Chaos. Pws Pub Co.