Which supplier delivers first?

Question

Supplier 1's delivery time is normally distributed with mean 50 days and standard deviation 5 days. Supplier 2's delivery time is normally distributed with mean 76 days and standard deviation 12 days. The two are independent.

What is the probability that Supplier 1 delivers before Supplier 2?

Accepted Answer

Compare the two times through their difference. Let D = (	ext{Supplier 1 time}) - (	ext{Supplier 2 time}). "Supplier 1 is faster" means it takes less time, that is, D < 0. The difference of two independent normals is normal: means subtract, variances add. Mean of D: \;50 - 76 = -26 days. Variance of D: \;5^2 + 12^2 = 25 + 144 = 169, so the standard deviation of D is \sqrt{169} = 13 days. So D is normal with mean -26 and standard deviation 13. Convert 0 to a z-score: z = \frac{0 - (-26)}{13} = 2. We want P(D < 0), the area to the left of z = 2, which is about 0.9772: P(D < 0) \approx 0.9772.…

Which supplier delivers first?

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