Mickey and Minnie plan to meet at a cafe, but each will independently arrive at a uniformly random time between 9:00 and 10:00. Mickey will wait 10 minutes for Minnie before leaving, and Minnie will wait 30 minutes for Mickey before leaving. What is the probability that they meet each other?

Free worked solution · Quant Blueprint

19/36, approximately 52.78%.

1. Measure both arrival times in minutes after nine. Their independent uniform arrivals give a 60 × 60 square of equally likely time pairs.
2. If Mickey arrives first, they fail when Minnie arrives more than ten minutes later. That region is a triangle with side length 50 and area 50²/2.
3. If Minnie arrives first, they fail when Mickey arrives more than thirty minutes later. That disjoint triangle has side length 30 and area 30²/2.
4. Subtract both failure areas: 1 − (1250+450)/3600 = 1900/3600 = 19/36.

Common mistake

Using one common waiting time erases the asymmetry. Waiting can extend past ten; only arrival times are restricted to the hour.

Follow-up

For an arrival window T and respective waits a,b in [0,T], P(meet) = 1 − [(T−a)²+(T−b)²]/(2T²).

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.