Bayes' Theorem implies that you can take the prior odds of the hypothesis A, or the ratio of its probability to the probability of its being false, A/a, and update that to take the evidence E into account by multiplying in the ratio of the probability of that evidence given A and given A: new odds = old odds * (E|A)/(E|a).

Play around with that until you see the truth of the claim you asked about. Note that A = 1-a.