Problem

You notice that at your subway station:

  • Train A arrives first 80% of the time
  • Train B arrives first only 20% of the time

Yet both trains are supposed to run with equal frequency

What’s going on?

Key Explanation

1. Frequency Mismatch?

It could be that Train A actually runs more frequently, despite what’s claimed. But that’s not the only explanation.

2. Uniform Arrival + Scheduling Offset

Even if both trains run every 10 minutes, you might still observe one coming first more often due to schedule offset.

Example:

  • Train A: 1:00, 1:10, 1:20, …
  • Train B: 1:02, 1:12, 1:22, …

Your arrival is uniformly random over time. Consider a typical 10-minute cycle between 1:00 and 1:10:

  • If you arrive between 1:00 and 1:02 (a 2-minute window), the next train is Train B (at 1:02).
  • If you arrive between 1:02 and 1:10 (an 8-minute window), the next train is Train A (at 1:10).

So:

\[ P(\text{Train A arrives first}) = \frac{8}{10} = \boxed{80\%} \]

Even though both trains arrive every 10 minutes!

Moral of the Story

Your random arrival time interacts with the train schedules in a way that biases your observation toward one train—even when both have equal frequency.

This is a real-world example of the inspection paradox or sampling bias due to scheduling structure.

Conclusion

The train that arrives first 80% of the time may not be more frequent—it just benefits from a favorable timing offset when you arrive randomly.

Reference