Define P(A) = probability that car A will start (the prior for car A starting)
P(B) = probability that car B will start (the prior for car B starting)
The probability that B starts given that A does not is the conditional probability P(B| ~A)
Given that P(B|~A) = .2 this implies that P(B|A) = .8 (since P(A) + P(~A) =1 )
Bayes rule states that the desired P(A|B) = P(B|A) P(A)/P(B)
To work out the ratio of priors ,P(A) /P(B), one must work with the other two givens.
These can be written as P(A) P(B) = .1 and (1-P(A)) (1 - P(B)) = .4
This set of two equations has two solutions: { P(A) = .5, P(B) = .2 } and {P(A) = .2, P(B) = .5 }
Thus the ratio of priors will be either .5/.2 = 2.5 or .2/.5 = .4
The first possibility must be rejected because Bayes rule would lead to P(B|A) = 2 which is larger than unity.
The second possibility leads to P(A|B) = .8 x .4 = .32
