All packages coming off a production line are weighed to determine if they are t

Question

All packages coming off a production line are weighed to determine if they are too light (i.e., underfilled). The scale identifies 96.9% of the underfilled packages as not meeting specifications, but it also (incorrectly) identifies 1.9% of acceptable packages as being underfilled. Based on historical studies of the process, it is believed that the actual percentage of all packages that are underfilled is 3.6%. The event U refers to a package actually being underfilled and U' refers to a package that is not underfilled. The event L refers to an item that the scale says is too light and L' refers to an item that the scale says is acceptable. We can summarize the given probabilities as follows: 0.969 = Probability the scale says a package is too light if it is underfilled 0.019 = Probability the scale says a package is too light if it is not underfilled 0.036 = Probability a package is underfilled a) Write each of the three probabilities above in terms of the events U, U', L, and L'. b) Given that the scale has identified a randomly selected package as being too light, what is the probability that the scale is correct and the package indeed is underfilled?

Explanation / Answer

A) P(U) = 0.036

P(U') = 1- 0.036 = 0.964

P(L) = 0.969x0.036 + 0.019x0.964 = 0.0532

P(L') = 0.036x(1-0.969) + 0.964(1-0.019) = 0.9468

B) P(underfilled | scale says light) = P(U | L)

= P(U and L)/ P(L)

= 0.969x0.036/0.0532 = 0.6557

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