The Birthday Paradox: Why Shared Birthdays Are More Common Than You Think

Ask a room of people how many are needed for a better-than-even chance that two share a birthday, and most guess well over a hundred. The real number is 23.

Why it's called a paradox

It isn't a contradiction, just a result that feels wrong. Our intuition asks, "what is the chance someone shares my birthday?", which is low. The actual question is whether any two people among the group share one — and there are far more pairs than people.

Counting pairs

In a group of 23, the number of different pairs is 23 × 22 ÷ 2 = 253. Each pair has roughly a 1 in 365 chance of matching, so with 253 chances the overall probability climbs quickly past 50 percent.

The calculation in outline

It is easier to compute the chance that nobody shares a birthday, then subtract from one. Assume 365 equally likely days. The first person can have any birthday. The second has 364 out of 365 choices to avoid a match, the third 363 out of 365, and so on. Multiplying these fractions for 23 people gives about 0.49, meaning a shared birthday has a probability of about 0.51.

How the probability grows

Approximate probabilities that at least two people share a birthday:

• 10 people: about 12 percent
• 23 people: about 51 percent
• 30 people: about 70 percent
• 50 people: about 97 percent
• 70 people: over 99.9 percent

Real-life caveats

The standard result assumes birthdays are spread evenly and ignores 29 February. Real births are not perfectly even — some months and days are more popular than others — which slightly increases the chance of matches. It is still a good approximation.

A party trick that works

At a gathering of 30 or more, or in a school class or football squad, you can bet a shared birthday exists and win more often than not. Sports squads of 23 (two teams of 11 plus a referee, in football) are a favourite example.

Where this idea appears elsewhere

The same mathematics underpins the "birthday attack" in cryptography, where collisions between random values occur far sooner than expected. It shows that a surprising number of everyday coincidences are not remarkable at all.

Bring it back to your own date

Once you know how likely shared birthdays are, look up exactly how many days separate yours from a friend's — the numbers make good party conversation.

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