I have a question which I found on the web:
Two boxes are placed in a cupboard out of which the first box contains 1 black and 3 red balls and the second box contains 4 black and 2 red balls. A ball picked at random from one of the boxes is found to be black. What is the probability that the ball is drawn from the first box?
I understand I have to use Bayes theorem for it, so something like $$Pr (First | Black) = \frac{Pr(Black|First) Pr(First)}{Pr(Black)}$$
Now my confusion is in how to compute $Pr(Black)$, the website calculates it using the sum of each individual marginals: $Pr(Black, First) + Pr(Black, Second)$. My question is why can't I just combine the two and say it is half? Since if we are going to be picking the box randomly as well as pick the ball randomly, so $Pr(Black) = \frac{1 + 4}{10}$, but that isn't right apparently?