Here we need to know about exponential distribution and bayes' theorem application..
Here P(Plant A) = 1 / 5
P(Plant B) = 4 / 5 [ The reason being plant B produces four times more than plant A ]
P(work after 12 months | plant A) = 1 - P(fails withing 12 months | plant A)
Now to calculate " P(fails withing 12 months | plant A) " ,
As it follows exponential distribution , here the parameter we use is θ and the expectation(mean) is 1 / θ ..
As 1/θ(mean) = 6 months (given) ==> θ = 1/6
Now P(fails within 12 months | plant A) = 1 - e-θa = 1 - e-1/6 . 12
= 1 - e-2
P(work after 12 months | plant A) = 1 - P(fails within 12 months | plant A)
= e-2
Similarly P(work after 12 months | plant B) = e-1/2.12
= e-6
Hence the inverse probability which is asked in question
i.e. P(plant A | work after 12 months) = P(plant A) * P(work after 12 months | plant A) / [ P(plant B) * P(work after 12 months | plant B) + P(plant A) * P(work after 12 months | plant A)
= 1/5 * e-2 / [ (1/5 * e-2 ) + (4/5 * e-6) ]
= 0.027 / (0.27 + 0.00198)
= 0.932
Hence required probability = 0.93 [correct to 2 decimal places]