Will our bid come out on top?

Question

Our sealed bid is normally distributed with mean 105 and standard deviation 3. A rival's bid is normally distributed with mean 100 and standard deviation 4. The two bids are independent, and the higher bid wins.

What is the probability that our bid beats the rival's?

Accepted Answer

Comparing two normal quantities is easiest through their difference. Let D = (	ext{our bid}) - (	ext{rival bid}). "We win" means our bid is higher, that is, D 0. The difference of two independent normals is itself normal. The mean subtracts, but the variances still add (spread always grows when you combine independent quantities, whether adding or subtracting): Mean of D: \;105 - 100 = 5. Variance of D: \;3^2 + 4^2 = 9 + 16 = 25, so the standard deviation of D is \sqrt{25} = 5. So D is normal with mean 5 and standard deviation 5. We want P(D 0). Convert 0 to a z-score on D: z = \frac{0 - 5}…

Will our bid come out on top?

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