# Hijacked probability thread **URL:** \ **Category:** Factual Questions\ **Created:** [April 1, 2001, 7:17pm UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397 "2001-04-01T19:17:28Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\ **Post date:** [April 1, 2001, 7:17pm UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/1 "2001-04-01T19:17:28Z")
I didn’t want to completely hijack a thread, so I moved it to a new one: here’s where we are now… > [@](#): > > [B}sethdallob > Member > > Registered: Apr 2000 > Posts: 379 > > ## quote: > > Originally posted by Cabbage > As for the OP, here’s an anology you may be familiar with–the old birthday problem. Ignoring leap days and assuming a uniform random distribution of birthdays, what’s the probability that, out of 23 people, at least two people have the same birthday (just month and day, not year)? It’s slightly over 50%, which is a lot higher than most people expect. > > * * * > > [hijack] Could you show me the math for that?? That is a potentially very lucrative bar bet.[/hijack] > > * * * > > -sethdallob > General Questions Manhattan > 04-01-2001 01:26 AM IP: Logged > > Cabbage > Member > > Registered: Sep 1999 > Posts: 1125 > Sure. The easiest way is to calulate the probability that no two people have the same birthday. Subtract that from one, and you’ve got the probability that at least two people have the same birthday. > > So if you have 23 people, the total number of different birthday distributions is 365^23 (each of the 23 people has 365 different possible birthdays). > > Now count the total number of different birthday distributions so that everybody has a different birthday. Take the people one at a time. The first person has 365 different possible birthdays. The second person has to have a different birthday from the first, so he only has 364 different possible birthdays. The third person’s birthday has to be different from the first two, so he has 363. The fourth has 362…The 23rd has 343. So there are > > 365 \* 364 \* 363 \* 362 \* … \* 343 > > birthday distributions among the 23 so that everybody has a different birthday. > > So the probability they all have different birthdays is: > > ## 365 \* 364 \* 363 \* 362 \* … \* 343 > > 365^23 > > = about 49.27% > > So the probability at least two have the same birthday is about 50.73%. > > So it’s just slightly over 50%. It gets larger pretty quickly. With 30 people, it’s 70.63%; for 35 it’s 81.44%; 40–89.12%; 45–94.10%; and for 50 there’s a 97.04% chance at least two will have the same birthday. So if you’re in a crowd of 45-50 people and can find someone to take the bet that two people will share a birthday, you’ve got a damn good chance of winning. > > * * * > > …ebius sig. This is a moebius sig. This is a mo… > (sig line courtesy of WallyM7) \*\* Pardon my skepticism, but this seems too good to be true. Does it really apply in the world (is the birthday pattern really even?)? I thought that there was a greater percentage of people born in Jan. - Mar., which could really blow up the odds and get expensive and embarrassing fairly quickly. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 1, 2001, 7:37pm UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/2 "2001-04-01T19:37:47Z")
> [@](#): > > I thought that there was a greater percentage of people born in Jan. - Mar. Actually, this is what’s sometimes referred to as the “Birthday Inequality”. If the birthday’s are truly randomly distributed, then you get the probabilities I mentioned above. If they’re not randomly distributed (more in Jan.-Mar., for example), it can only _increase_ the chances of at least two sharing a birthday, so you’d have an even _better_ chance of winning a bet, in that case. ---
**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\ **Post date:** [April 1, 2001, 8:43pm UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/3 "2001-04-01T20:43:47Z")
To test your theory, I went through my PalmPilot’s database, which has 72 ppl. in it, all with different birthdays. Either I’m one of the 1%, or something is amiss. Please note I’m just playing devil’s advocate here; something just smells funny, and before I lose hundreds of dollars I like to make sure :). ---
**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\ **Post date:** [April 1, 2001, 10:26pm UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/4 "2001-04-01T22:26:36Z")
I think you are using probability for large numbers on small ones - thus your accuracy coulb be way off. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 2, 2001, 1:55am UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/5 "2001-04-02T01:55:05Z")
With 72 people, it’s 99.95% likely that at least two will have the same birthday. Did you choose the 72 people randomly, or did you intentionally pick 72 that all have different birthdays? If you picked them randomly and they all have different birthdays, all I can say is that’s a huge fluke; there’s only a 1 in 2000 chance of that happening. I can assure you that this isn’t just some quirk of the math, though, it really does “work” in the real world. Here’s some more about it if you’re still skeptical, and from there are additional links to even more pages, as well: [http://forum.swarthmore.edu/dr.math/faq/faq.birthdayprob.html](http://forum.swarthmore.edu/dr.math/faq/faq.birthdayprob.html) ---
**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\ **Post date:** [April 2, 2001, 2:04am UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/6 "2001-04-02T02:04:22Z")
I humbly submit the following article, and wait for your opinion: [http://www.maa.org/devlin/devlin\_4\_00.html](http://www.maa.org/devlin/devlin_4_00.html) My reading seems to say you are incorrect. However, I’m tired, and I’ve been thinking about other issues today. 🙂 ---
**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\ **Post date:** [April 2, 2001, 2:06am UTC](https://boards.straightdope.com/t/hijacked-probability-thread/60397/7 "2001-04-02T02:06:18Z")
Now I’ve re-read it, and it appears you are right. Sorry :o