Three routes to work with different travel times

Question

Each morning you pick one of three routes to work. You take the highway with probability $0.5$ (average time $20$ minutes), the back roads with probability $0.3$ (average $30$ minutes), or the scenic route with probability $0.2$ (average $45$ minutes).

How long should you expect your commute to take?

Accepted Answer

Two stages: which route you choose, then the travel time on that route. The tower rule blends the route averages by how often you take each: E[	ext{time}] = E\big[\,E[	ext{time} \mid 	ext{route}]\,\big]. Step 1, the route averages are given: highway 20, back roads 30, scenic 45. Step 2, blend by the route probabilities: E[	ext{time}] = 0.5 	imes 20 + 0.3 	imes 30 + 0.2 	imes 45 = 10 + 9 + 9 = 28 	ext{ minutes}. You expect a 28 minute commute. The scenic route is slow but rare, so it adds only 9 minutes to the blend. Takeaway. With more than two sources the recipe is unchanged: weight …

Three routes to work with different travel times

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