The Office Bet

How it works

Twenty-three people are in a meeting room. Dev puts fifty pounds on the table and bets that at least two of them share a birthday. Birthdays are uniform over 365 days and independent, and Dev wins if any two people match - not if someone matches him. The story asks whether you take the bet. You lock in a guess, then step through the answer and breakdown.

How scoring works

No grading - your guess is displayed back beside the answer. The story teasers share one lifetime unlock for free users.

The key insight

Compute the complement: the chance all 23 birthdays differ is (365/365)(364/365)...(343/365), about 0.493, so at least one match happens about 50.7% of the time - Dev is a slight favourite, and at 22 people he would have been a slight underdog. Intuition fails because matches happen between pairs, and 23 people contain C(23,2) = 253 pairs, not 23 chances.

The full step-by-step solution lives at /brain-teasers/birthday-problem/solution.

A worked example

Let people into the room one at a time: the second must dodge 1 day, the third 2, down to the 23rd dodging 22. Multiplying the chain gives about 0.4927 for all-distinct, so P(match) is about 50.7%. Don't take the bet.

Common mistakes

• Comparing 23 people to 365 days - the bet is about pairs, which grow quadratically.

• Confusing 'anyone matches anyone' with 'someone matches me' - the latter is 22 comparisons and sits around 6%.

• Attacking 'at least one' directly instead of computing the complement.

Why interviews test this

Interviewers are checking two habits - reaching for the complement, and knowing pair-counts scale quadratically - both of which transfer straight to collision estimates. The written solution is at /brain-teasers/birthday-problem/solution.