Birthday Paradox & Monty Hall Calculator
How likely is a shared birthday among 23 people? (50.7%) Should you switch doors in Monty Hall? (Yes — 2/3) Verified analytic values with explanations.
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Match probability among 23 people
50.7297%
- Number of pairs
- 253
How it was calculated
- 01Probability all birthdays differ
365 × 364 × … × 343 ÷ 365^23
Basis: Complement via the multiplication rule — probability every pair differs
- 02Probability of at least one matching pair
1 − 0.492703 = 50.73%
- 03Why does intuition fail?
The question isn't 'someone shares MY birthday' but 'ANY two of n people match'. Possible pairs grow as n(n−1)/2 — at 23 people that's already 253 pairs, crossing 50%.
Reference simulation only, not tax advice. Verify with official sources before filing or paying.