The Setup

Two people (say, bankers) independently arrive at a train station at a time uniformly at random between 5:00 AM and 6:00 AM. Upon arrival, each waits exactly 5 minutes before departing.

Question: What is the probability that their waiting intervals overlap so that they actually meet?

Visualizing the Problem

Let:

  • \( X \): arrival time of person A (in minutes after 5:00 AM)
  • \( Y \): arrival time of person B (also in minutes after 5:00 AM)

Each variable \( X, Y \) is uniformly distributed on the interval \( [0, 60] \).

They meet if their arrival times are within 5 minutes of each other:

\[ |X - Y| \leq 5 \]

Geometric Probability Approach

We represent all possible arrival pairs \( (X, Y) \) as points in the 60×60 square.

The region where \( X - Y \leq 5 \) forms a band of width 10 along the diagonal.

Total Area:

\[ 60 \times 60 = 3600 \]

Favorable Area (meeting zone):

The area where they do not meet is where \( X - Y > 5 \). This region consists of two right triangles, each with a base and height of 55.

\[ \text{Area outside band} = 2 \times \frac{1}{2}(55)^2 = 3025 \]

\[ \text{Area where } |X - Y| \leq 5 = 3600 - 3025 = 575 \]

So:

\[ P(\text{meeting}) = \frac{\text{favorable area}}{\text{total area}} = \frac{575}{3600} = \boxed{\frac{23}{144}} \approx 0.1597 \]

Final Answer

\[ \boxed{\frac{23}{144}} \]

Reference