how to calculate discrete event probability? A bookstore sells children's books that belong to two publishing companies A and B and were published between years 2000 and 2004. The probabilities of a book being published by companies A and B are 0.6 and 0.4. The probability that company A published a book in years 2000, 2001, 2002, 2003, 2004 are 0.1, 0.2, 0.3, 0.3 and 0.1 respectively. The probability that company B published a book in years 2000, 2001, 2002, 2003, 2004 are 0.3, 0.2, 0.1, 0.2 and 0.2 respectively.
a) Find the probability that a book was published after 2002
b) Find the probability that a book published after 2002 was published by company A.
I think that for solution a)
probability that a book was published after 2002 is
$$\begin{align}
P(A \cup B) &= P(A) + P(B) - P(A)P(B)
\end{align}
$$
which is equal to
$$\begin{align}
P(A \cup B) &= \frac{4}{10}+\frac{4}{10}-\frac{4}{10}\frac{4}{10} = 0.64
\end{align}
$$
for solution b) i don't understand the question exactly but 
If I assume that it asks only the probability that a book published after 2002 was published by company A
$$\begin{align}
P(A \cup \overline{B}) &= P(A) + P(\overline{B}) - P(A)P(\overline{B})
\end{align}
$$
which is equal to
$$\begin{align}
P(A \cup B) &= \frac{4}{10}+\frac{6}{10}-\frac{4}{10}\frac{6}{10} = 0.76
\end{align}
$$
Otherwise if I assume that only the probability that a book published after 2002 was published by company A and the probability that a book published after 2002 was published by company A and B
So the solution is
$$
P(A) = 4/10 = 0.4
$$
Is solution of a is correct and for b which solution is correct?
Thanks in advance.

Do you see how you can convert all the probabilities to these values?
And does that make it easier to figure out the answers?
$$\begin{array}{|c|c|c|c|c|c|} \hline
\text{Company}& \text{2000}& \text{2001}& \text{2002}& \text{2003}& \text{2004} \\ \hline
\text{A} & .06& .12& .18& .18& .06  \\ \hline
\text{B} & .12& .08& .04& .08& .08  \\ \hline
\end{array}$$
There is no "published by A and B".  They are mutually exclusive.  So to find part b), you need to find what percentage of books published after 2002 were the ones published by Company A.
