Question 875108
Events:


C: item possesses a certain characteristic
C': item does not posses a certain characteristic
D: item has a defect
D': item does not have a defect



Probabilities:


P(C): probability that you choose an item that possesses a certain characteristic
P(C'): probability that you choose an item that does not possess a certain characteristic
P(D): probability that you choose an item that is defective
P(D'): probability that you choose an item that is not defective


P(C|D): probability that you choose an item that possesses a certain characteristic given the item has a defect
P(C'|D): probability that you choose an item that does not possess a certain characteristic given the item has a defect
P(C|D'): probability that you choose an item that possesses a certain characteristic given the item does not have a defect
P(C'|D'): probability that you choose an item that does not possess a certain characteristic given the item does not have a defect


P(D|C): probability that you choose an item that is defective given the item has that certain characteristic
P(D'|C): probability that you choose an item that is not defective given the item has that certain characteristic
P(D|C'): probability that you choose an item that is defective given the item does not have a certain characteristic
P(D'|C'): probability that you choose an item that is not defective given the item does not have a certain characteristic


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Given Probabilities


"10% of the units in a batch have a common defect"
P(D) = 0.10
P(D') = 0.90 (1 - 0.10 = 0.90)



"Experience show that 87% of the defective units posses a certain characteristic"
P(C|D) = 0.87 (given a defective unit, 87% of them will have this characteristic)


"only 3% of the units which do not have this defect posses that characteristic"
P(C|D') = 0.03 (given a non-defective unit, 3% of them will have this characteristic)


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"A unit is examined and found to have the characteristic. What's the conditional probability that the unit has the defect?" so we want to find P(D|C). This is the probability of event D happening given event C has happened.



We want to find P(D|C)


Use <a href="http://en.wikipedia.org/wiki/Bayes'_theorem">Baye's Theorem</a> to get


P(D|C) = (P(C|D)*P(D))/(P(C))



Then use the <a href="http://en.wikipedia.org/wiki/Law_of_total_probability">Law of Total Probability</a> to break up P(C) into P(C|D)*P(D)+P(C|D')*P(D')



We will now have this equation


P(D|C) = (P(C|D)*P(D))/(P(C|D)*P(D)+P(C|D')*P(D'))



Plug in the values: P(D) = 0.10, P(D') = 0.90, P(C|D) = 0.87, P(C|D') = 0.03 and evaluate


P(D|C) = (P(C|D)*P(D))/(P(C|D)*P(D)+P(C|D')*P(D'))
P(D|C) = (0.87*0.10)/(0.87*0.10+0.03*0.90)
P(D|C) = <font color="red">0.76315789473684</font>