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Question 313:



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You'd want to use Bayes' Theorem here. We have some conditional probabilities and need to work backwards.  We know a customer failed to learn the skill, so we want the conditional probability of A given failure, or written formally, P(A| F). Part of the Baysian theorem states that the conditional probability of A given F = the probability A * probability of F divided by the probability of F.

So first lets find the probability the customer failed. We'd use the addition rule and add up all the ways to fail times the probability of each failure for methods A, B & C.

 .2(.40) + .1(.10)  +.3(.50)  = .24  probability of failure.

Notice that the probability of A being selected wasn't stated, since there are only 1 of 3 ways, A has to be 1- the probability of B and C, 1-(.1+.5) = .4.

Now that we have the denominator, we find the numerator as the probability of A = P(A) = .4 times the probability of failure for A which is .2*.40 = .08.

The equation is  .08 / .24 = .333 or the probability the failed customer took method A was about 33.3%. 

We can just plug in the probability of failure for B and C as well: .01/.24 and .15/.24 = .042 and .625 repectively. So given that we've observed a failured, the chanes are the customer learned method A as 33.3%, B as 4.2% and C as 62.5%. As a final check, all probabilities add up to one (.333+.042+.625) =1.

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