How to Calculate Birthday Paradox
What is Birthday Paradox?
The Birthday Paradox is a famous probability result: in a group of just 23 people, there is a greater than 50% probability that two people share a birthday. This surprises most people because 23 seems small compared to 365 days. The probability grows rapidly — with 70 people it exceeds 99.9%. It is called a paradox not because it is logically contradictory, but because it strongly violates intuition.
Formula
- P
- 1 − ∏ᵢ₌₀ⁿ⁻¹ (365−i)/365 — 1 − ∏ᵢ₌₀ⁿ⁻¹ (365−i)/365
Step-by-Step Guide
- 1P(at least one shared birthday) = 1 − P(all different)
- 2P(all different) = (365/365) × (364/365) × (363/365) × ... × ((365−n+1)/365)
- 3P(match) = 1 − ∏ᵢ₌₀ⁿ⁻¹ (365−i)/365
- 4Assumes uniform birthday distribution (actual distribution varies slightly)
Worked Examples
Frequently Asked Questions
What is Birthday Paradox?
The Birthday Paradox is a famous probability result: in a group of just 23 people, there is a greater than 50% probability that two people share a birthday. This surprises most people because 23 seems small compared to 365 days
How accurate is the Birthday Paradox calculator?
The calculator uses the standard published formula for birthday paradox. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.
What units does the Birthday Paradox calculator use?
This calculator works with inches, percentages. You can enter values in the units shown — the calculator handles all conversions internally.
What formula does the Birthday Paradox calculator use?
The core formula is: P(at least one shared birthday) = 1 − P(all different). Each step in the calculation is shown so you can verify the result manually.
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