You were late, but which route did you take?

Question

Route X makes you late 20\% of the time. Route Y is slower and makes you late 50\% of the time. Each morning you pick a route by flipping a fair coin. Today you arrived late. What is the probability you took Route X?

Accepted Answer

Both routes are equally likely at the start. Imagine 100 mornings, about 50 on each route: - Route X: 50 	imes 0.20 = 10 late arrivals. - Route Y: 50 	imes 0.50 = 25 late arrivals. Of the 10 + 25 = 35 late mornings, only 10 used Route X: P(	ext{X} \mid 	ext{late}) = \frac{10}{35} = \frac{2}{7} \approx 0.286. Being late is a stronger signal of Route Y, so the evidence pulls your belief down from 50 percent to about 29 percent for Route X. Takeaway. Evidence that a cause rarely produces (lateness from the reliable route) makes that cause less likely once you condition on it.

You were late, but which route did you take?

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