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The Birthday Paradox

In a room of just 23 people, two probably share a birthday. Not a paradox — just more pairs than intuition expects, and a complement that is easy to compute.

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Ten people in a room. The chance that some two of them share a birthday is 11.7% — the curve at n = 10. The dots are 500 simulated rooms for each size.

P(\text{shared}) = 1 - \tfrac{365}{365} \cdot \tfrac{364}{365} \cdot \tfrac{363}{365} \cdots \tfrac{356}{365} = 1 - 0.8831 = \mathbf{0.1169} \\[6pt] \binom{10}{2} = 45 \text{ pairs}, \qquad 1 - e^{-45/365} = 0.116

People in the room10Simulated rooms per dot500

The Birthday Paradox

In a room of just 23 people, two probably share a birthday. Not a paradox — just more pairs than intuition expects, and a complement that is easy to compute.

Step 1 of 7

Speed

Ten people in a room. The chance that some two of them share a birthday is 11.7% — the curve at n = 10. The dots are 500 simulated rooms for each size.

P(\text{shared}) = 1 - \tfrac{365}{365} \cdot \tfrac{364}{365} \cdot \tfrac{363}{365} \cdots \tfrac{356}{365} = 1 - 0.8831 = \mathbf{0.1169} \\[6pt] \binom{10}{2} = 45 \text{ pairs}, \qquad 1 - e^{-45/365} = 0.116

People in the room10Simulated rooms per dot500

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The Birthday Paradox

See also

The Birthday Paradox

See also