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## Homework Statement

A chemical engineer is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of 0.80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is 0.90. The prior probabilities of the impurity being present and being absent are 0.40 and 0.60 respectively. Three separate experiments result in only two decisions. What is the posterior probability that the impurity is present?

## Homework Equations

## The Attempt at a Solution

So I know that the event that an impurity is present and not present are disjoint and exhaustive, so Bayes Theorem does apply. I let [itex] D [/itex] denote the event that an impurity was detected in 2 of 3 tests and [itex] I [/itex] denotes the event that an impurity is present.

[itex] \mathcal P{(I|D)} = \frac{\mathcal P{(D|I)}* \mathcal P{(I)}}{\mathcal P{(D|I)}* \mathcal P{(I)} + \mathcal P{(D|I')}* \mathcal P{(I')}} [/itex]

Also, I know that the probabilities for [itex] \mathcal P{(I)} = 0.40 [/itex] and [itex] \mathcal P{(I')} = 0.60 [/itex]

However, at this point I am not sure where to go with the problem. I understand that I need to determine [itex] \mathcal P{(D|I)} [/itex] and [itex] \mathcal P{(D|I')} [/itex] but I'm not sure how to go about doing this..