It seems to me that the distribution of the number of defects per 100 feet is missing. I'm guessing they want you to think of it as a Poisson process. If X is the number of defects in the 1200 foot thing, then we should have X|theta ~ Poisson(12theta). So, you want to get the distribution of theta|X, evaluated at X = 4. Use the fact that

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You should get a gamma for the posterior I think.