An author writes a good book with a probability of 1/2. If it is good 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 write two.

The problem can be solved by considering the different scenarios in which at least one book gets published.

1. Both books are good and both get published.
2. Both books are good, but only one gets published.
3. One book is good and gets published, the other is not good but gets published.
4. One book is good and gets published, the other is not good and does not get published.
5. One book is not good but gets published, the other book is good but does not get published.
6. Both books are not good, but one gets published.

We calculate the probability for each scenario and sum them up.

1. The probability that both books are good is (1/2) * (1/2) = 1/4. The probability that both get published is (2/3) * (2/3) = 4/9. So, the total probability for scenario 1 is (1/4) * (4/9) = 1/9.

2. The probability that both books are good is still 1/4. The probability that only one gets published is 2 * (2/3) * (1 - 2/3) = 4/9. So, the total probability for scenario 2 is (1/4) * (4/9) = 1/9.

3. The probability that one book is good and the other is not is 2 * (1/2) * (1 - 1/2) = 1/2. The probability that the good one gets published and the not good one also gets published is (2/3) * (1/4) = 1/6. So, the total probability for scenario 3 is (1/2) * (1/6) = 1/12.

4. The probability that one book is good and the other is not is still 1/2. The probability that the good one gets published and the not good one does not get published is (2/3) * (1 - 1/4) = 1/2. So, the total probability for scenario 4 is (1/2) * (1/2) = 1/4.

5. The probability that one book is not good and the other is good is still 1/2. The probability that the not good one gets published and the good one does not get published is (1/4) * (1 - 2/3) = 1/12. So, the total probability for scenario 5 is (1/2) * (1/12) = 1/24.

6. The probability that both books are not good is (1 - 1/2) * (1 - 1/2) = 1/4. The probability that one gets published is 2 * (1/4) * (1 - 1/4) = 3/8. So, the total probability for scenario 6 is (1/4) * (3/8) = 3/32.

Adding up all these probabilities gives the total probability that at least one book gets published:

1/9 + 1/9 + 1/12 + 1/4 + 1/24 + 3/32 = 0.7083 or approximately 71%.