Talk:Birthday problem

logarithms and combinations

I just noticed this, which I don't see in the article: ${\frac {\log(1/2)}{\log(364.25/365.25)}}\approx 252.83\approx {\binom {23}{2}}$ —Tamfang (talk) 18:01, 28 March 2024 (UTC)

So what? --Macrakis (talk) 20:37, 28 March 2024 (UTC)

364.25/365.25 is the probability that a given pair do not share a birthday. 253 is the number of pairs among 23 people. I never knew before why the threshold number is 23. —Tamfang (talk) 06:15, 29 March 2024 (UTC)

So you're saying that this is more than a coincidence? --Macrakis (talk) 15:45, 29 March 2024 (UTC)

Much closer, but equally meaningless: 365*log(2) = 252.999. --Macrakis (talk) 21:17, 28 March 2024 (UTC)

Or to put that another way, 23 is the smallest integer n such that $({\frac {364.25}{365.25}})^{\binom {n}{2}}\leq {\frac {1}{2}}$. —Tamfang (talk) 00:03, 30 March 2024 (UTC)

OK, I think I'm beginning to follow you here. Small detail: article says that leap years aren't taken into account, so it should be 364/365. --Macrakis (talk) 15:08, 1 April 2024 (UTC)

Taking that further, the approximation n >= 0,5 + sqrt(0,25 + 2 * ln(2) * 365) mentioned in the article could then be improved by writing it as n >= 0,5 + sqrt(0,25 + 2 * ln(2) / ln (365/364)) Roelgrif (talk) 02:16, 11 January 2025 (UTC)

Not related to real life

There are no examples here of real life tests. Like going into a classroom or office and asking people. If this is just a numbers game it should be made clear at the start. 2A00:23C8:E414:1F01:6D85:5817:D045:A7F3 (talk) 23:16, 10 April 2025 (UTC)

The article makes clear that it's about probability. The article also later mentions real-world probability due to unneven distribution.  …/NemoThorx (💬 • 📜) 11:35, 4 August 2025 (UTC)

Any real life test would be original research. The point is that it doesn't need a real life test, because it's a mathematical concept. All you need is a piece of chalk.User:Pedant (talk) User:Pedant (talk) 03:13, 10 January 2026 (UTC)

interactive widget

Try the birthday problem

Group size: 23 23

0%50%100%

In a group of 23 people, the probability that at least two share a birthday is 50.7%.

A shared birthday is more likely than not!

Deal random rooms

How often does a random room of that size really contain a match?

Deal a room of 23 people

Deal a room to see the result.

No room dealt yet - the boxes fill when you deal.

Jan 1 Jan 2 Jan 3 Jan 4 Jan 5 Jan 6 Jan 7 Jan 8 Jan 9 Jan 10 Jan 11 Jan 12 Jan 13 Jan 14 Jan 15 Jan 16 Jan 17 Jan 18 Jan 19 Jan 20 Jan 21 Jan 22 Jan 23 Jan 24 Jan 25 Jan 26 Jan 27 Jan 28 Jan 29 Jan 30 Jan 31 Feb 1 Feb 2 Feb 3 Feb 4 Feb 5 Feb 6 Feb 7 Feb 8 Feb 9

Each box is one person - a birthday drawn uniformly from 365 days, never re-rolled to force a result. Slide the group size to grow or shrink the room.

Long-run tally

Every deal of the same group size adds to this tally. Changing the slider starts a fresh tally on the next deal.

Rooms of 23	With a match	Match %	Expected
0	0	0.0	50.7%

▶ Auto-deal

Reset

50.7 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 23 1 50.7

The birthday problem at a glance (365 equally likely birthdays)

People in the group	Chance of a shared birthday
5	2.7%
10	11.7%
23	50.7%
30	70.6%
50	97.0%
70	99.9%

With just 23 people, a shared birthday is already more likely than not. An interactive version of this demonstration is available with JavaScript enabled.

Hi all. I've been on an interactivity kick lately, given the possibilities opened by the new {{calculator}} template, and spent some time figuring out an interactive version of the birthday problem for the article. Disclosure: I used Claude Code to make the calculator syntax more manageable. This one's been through several versions at this point, trying to make it the most useful. I still have ambitions to replace the random dates with a random list of notable people (see this VPT thread), but it's usable as-is. What do we think? Worth adding? — Rhododendrites ^talk \\ 22:45, 30 July 2026 (UTC)