Mathematics
The Birthday Paradox: Probability in a Shared Calendar
Quick fact
With only 23 people, the probability of a shared birthday is about 50.73% — and with 70 people, it jumps to over 99.9%.
Why this is interesting
If you’re in a room with just 22 other people, there’s a greater than 50% chance that at least two of you share a birthday. How can that be?
Read the full explanation
Understanding The Birthday Paradox: Probability in a Shared Calendar
Most people imagine comparing one person to all others, which seems to require many people to reach 50%. But the paradox works because every pair is compared: with 23 people, there are 253 possible pairs (23 × 22 / 2). When you check all pairs, the chance of at least one match grows much faster than our intuition expects.
A deeper explanation
The key is to calculate the complement: the probability that all birthdays are different. For 23 people, that’s (365/365) × (364/365) × (363/365) × … × (343/365). Multiplying these gives about 0.4927, so the probability of at least one match is 1 – 0.4927 = 0.5073. This multiplications of decreasing factors reveals how quickly the chance of a match climbs, and the same principle applies to any collision problem, such as two files having the same hash value.