# Birthday Probabilities **URL:** \ **Category:** Factual Questions\ **Created:** [May 17, 2010, 4:21pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776 "2010-05-17T16:21:12Z")\ **Posts on this page:** 6\ **Page:** 1
**Author:** ![MatthewGerlach](https://avatars.discourse-cdn.com/v4/letter/m/5e9695/32.png) [@MatthewGerlach](https://boards.straightdope.com/u/MatthewGerlach)\ **Post date:** [May 17, 2010, 4:21pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776/1 "2010-05-17T16:21:12Z")
Today I was approached with a ‘strange question’. I was told that in a class of 30 students, the probability that any two will have their birthdays on the same day (of the same month) is of 89%. At first, it seemed absurd, and after some quick calculations I can’t get even close to 89%. However, the person who told me of this continues persistent and insists that she is right. Can somebody help out? Thanks. ---
**Author:** ![Leaffan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leaffan/32/299_2.png) [@Leaffan](https://boards.straightdope.com/u/Leaffan)\ **Post date:** [May 17, 2010, 4:23pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776/2 "2010-05-17T16:23:59Z")
> **[Birthday problem](https://en.wikipedia.org/wiki/Birthday_problem)** > > In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share a birthday. The birthday paradox refers to the counterintuitive fact that only 23 people are needed for that probability to exceed 50%. > The birthday paradox is a veridical paradox: it seems wrong at first glance but is, in fact, true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result ... ---
**Author:** ![Colibri](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colibri/32/1841_2.png) [@Colibri](https://boards.straightdope.com/u/Colibri)\ **Post date:** [May 17, 2010, 4:53pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776/3 "2010-05-17T16:53:40Z")
Note that the probability refers not to “any two” having the same birthday, but that in a room of 30 people, at least one pair of people will have the same birthday (on no particular date). ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [May 17, 2010, 4:56pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776/4 "2010-05-17T16:56:08Z")
From **Leaffan** ’s link, it’s only about 71%. The 89% figure in the OP is correct for 40 students, not 30. Maybe that’s the disconnect. ---
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [May 17, 2010, 7:49pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776/5 "2010-05-17T19:49:55Z")
For any group, how many possible pairs are there? i.e. for 5 people, we have 10 pairs: 1-2, 1-3, 1-4, 1-5 2-3, 2-4, 2-5 3-4, 3-5 4-5 Basically, n(n-1)/2 pairs; since 1-1, 2-2, etc are meaningless and e.g. 1-2 is same as 2-1. So for 30 people, there are 30\*29/2= 435 pairs. The odds that one of those pairings has a common birthday? Seems pretty good. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [May 17, 2010, 8:04pm UTC](https://boards.straightdope.com/t/birthday-probabilities/539776/6 "2010-05-17T20:04:09Z")
I think this Birthday Problem is commonly misunderstood, like MatthewGerlach’s friend did. People are surprised to hear that the probability is very high, but assume some different question such as that it applies to “any two” people. Once you know what the probability is for, the answer isn’t so surprising.