A quick two-leg commute

Question

A commute has two independent legs. The first leg takes a normally distributed time with mean 40 minutes and standard deviation 6 minutes. The second leg has mean 20 minutes and standard deviation 8 minutes.

What is the probability that the whole commute takes less than 45 minutes?

Accepted Answer

Adding two independent normal times gives another normal time. The means add directly, and the variances (squared standard deviations) add. Mean of the total: 40 + 20 = 60 	ext{ minutes}. Variance of the total: 6^2 + 8^2 = 36 + 64 = 100, so the total's standard deviation is \sqrt{100} = 10 minutes. The commute time is normal with mean 60 minutes and standard deviation 10 minutes. Convert the 45-minute target to a z-score: z = \frac{45 - 60}{10} = -1.5. Since we want "less than," we take the area to the left of z = -1.5, which is about 0.0668: P(	ext{total} < 45) \approx 0.0668. A commute th…

A quick two-leg commute

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