Probability Two Bankers Meet at the Station

Two people arrive independently and uniformly at random between 5:00 AM and 6:00 AM. Each person waits exactly 5 minutes. What is the probability that their waiting intervals overlap, so they meet?

Geometric Visualization

5:00 5:10 5:20 5:30 5:40 5:50 6:00 5:00 5:10 5:20 5:30 5:40 5:50 6:00 Banker B arrival time Banker A arrival time MEET REGION |A − B| ≤ 5 Too far apart Too far apart

The square represents all possible arrival pairs \((x,y)\), where each coordinate is between 0 and 60 minutes. They meet exactly when the difference in arrival times is at most 5 minutes.

Key Idea

Let x and y be the two arrival times, measured in minutes after 5:00.
They meet if their 5-minute waiting intervals overlap:
|x - y| ≤ 5
Total sample space = all points in a 60 × 60 square:
60² = 3600
The failure region consists of two congruent right triangles, each with leg length 55:
Area of one triangle = (1/2) × 55 × 55 = 1512.5

Step-by-Step Calculation

1. Total area:

A_total = 60 × 60 = 3600

2. Area where they do not meet:

This happens when one banker arrives more than 5 minutes before the other:

|x - y| > 5

That creates two triangles, each with side length \(60 - 5 = 55\).

A_fail = 2 × (1/2 × 55 × 55) = 55² = 3025

3. Area where they meet:

A_meet = 3600 - 3025 = 575

4. Probability:

P(meet) = 575 / 3600 = 23 / 144 ≈ 0.1597
Answer: 23/144 ≈ 15.97%

Interpretation

  • If they arrive within 5 minutes of each other, they overlap and meet.
  • If they are more than 5 minutes apart, one leaves before the other arrives.
  • Geometrically, this is the diagonal strip around the line y = x.

Monte Carlo Check

Click to simulate many random arrival pairs.

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A useful interview trick: whenever two random times need to be “close enough,” map them to a square and look for a diagonal band.