Curiosities · Foundation · 7 min

The birthday paradox

A small room contains far more possible pairs than people.

WHAT HAPPENS IF…Do 23 people make a shared birthday more likely than not?

Any shared birthday50.73%

Possible pairs253

All birthdays different49.27%

Exact theoretical probability under independent, uniformly distributed birthdays across 365 days; no leap day. Pair events are not independent, so we compute the complement of all birthdays being distinct.

Preparing your experiment…

Connect the picture to the idea

P(\text{shared})=1-\prod_{k=0}^{n-1}\left(1-\frac{k}{365}\right)

Compute the complement. The first person is unrestricted. The next avoids one occupied date, the next avoids two, and so on. Multiplying these conditional probabilities gives the chance of all birthdays being different. Subtracting from one gives the chance of at least one match.

A detail that matters

The pair count is an intuition, not an independence argument: pair-sharing events are dependent. Real birthdays are not exactly uniform.

A bridge to another ideaAnother probability that surprises ↗Careful conditioning can overturn an intuitive first guess.

Keep the connection going

One idea leads to another.

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