Why 23 people is enough to share a birthday
It feels like it should take hundreds. It takes 23 — and the reason is that you are not looking for someone who shares YOUR birthday, you are looking for any pair at all.
Why 23 people is enough to share a birthday — the interactive part
The first 20 trials, one square each
10 / 20
counteddid not
Share of groups with a shared birthday
People in the room
This chart as text
The exact curve runs from near 0.0% at 1 to 100.0% at 80. At 23 the simulation measured 50.7% against an exact 50.7%.
Runs a fresh set of trials. Moving a slider keeps the same random draws, so only the thing you changed changes.
In 5,000 simulated groups is 50.7%, 2,533 had a match. Exact probability is 50.7%, 23 people is the 50/50 point. Simulation is off by is 0.07 percentage points.
You are counting the wrong thing
The instinct is to compare 23 against 365 and conclude it cannot possibly be enough. But nobody is looking for a match with one particular person. Twenty-three people make 253 different pairs, and every one of those pairs is another chance. The question is not 'how many people' but 'how many pairs', and pairs grow roughly with the square of the group.
What the simulation adds
The exact answer needs no simulation: multiply together the shrinking chances that each new person misses every birthday already taken. The simulation earns its place by showing the answer is not a trick of the algebra — actually deal out thousands of rooms of people and roughly half of them really do contain a match. Watch the estimate settle as the trial count grows.
Check your understanding
How many people before a shared birthday is essentially certain — above 99%?
Pick one to check yourself.