Probability of finding the answer with multiple cases

Question

An author writes a good book with a probability of 1/2. If it is good it is published with a probability of 2/3. If it is not, it is published with a probability of 1/4. Find the probability that he will get atleast one book published if he writes two.

My approach $\ P_{Good}=\frac{1}{2}$&$\ P_{Bad}=\frac{1}{2}$

$\ P_{Good-Published}=\frac{2}{3}$&$\ P_{Good-NotPublished}=\frac{1}{3}$

$\ P_{Bad-Published}=\frac{1}{4}$&$\ P_{Bad-NotPublished}=\frac{3}{4}$

The author writes two books and one needs to be published. The answer is $\frac{407}{576}$

I am not able to get the answer.

Answer 1

At the first step, let to compute the probability of publishing a book. $$p= P(\text{publish a book}) = P(Good)\times P(Publish|Good) + P(Bad)\times P(Publish|Bad) = \frac{1}{2}\times \frac{2}{3} + \frac{1}{2}\times \frac{1}{4}$$

We know publishing a book is independent of publishing another book. Thus, we have $$P(\text{publish at least one book}) = 1- P(\text{does not publish any book}) = 1- (1-p)^{2} = \frac{407}{576}$$