The birthday paradox: how many people until two share a birthday?

The riddle that always surprises

How many people do you need in a room to have a more than 50% chance that two of them share a birthday?

Instinctively, you think 183 (half of 365). The real answer: 23.

The curve that goes « oh »

Some striking values:

• 10 people → ≈ 12%
• 23 people → ≈ 50.7% (the famous threshold)
• 50 people → ≈ 97%
• 70 people → ≈ 99.9%

Why it’s not 183

The classic mistake: comparing yourself to others. In fact, you have to count all possible pairs.

With $n$ people, the number of pairs is $\frac{n(n-1)}{2}$. With 23 people, that’s 253 pairs — each with a small chance (~1/365) of collision. 253 × (1/365) is already near 0.7. The number of pairs grows fast with $n$.

The exact formula

The probability of no collision:

$P(\text{none}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \dots \times \frac{365 - n + 1}{365}$

And:

$P(\text{at least one collision}) = 1 - P(\text{none})$

The lesson to remember

Our intuition underestimates how fast pairs multiply. It’s the same trap in many other probability problems: count pairs (or groups), not individuals.

?Your turn

In a class of 30 students, the probability that two share a birthday is roughly:

A.3%B.12%C.70%D.50%

Check