Having this situation
http://i.stack.imgur.com/PE226.jpg
two urns with the number of balls in there pictured above..
and two events
A = urn is 1 B = ball is white
I know that $P(A) = 1/2$, $P(\text{not }A) = 1/2$, $P(B \mid A) = 2/3$, $P(B\mid\text{not }A) = 3/4$
but if I try to verify the Bayes Theorem with P(B/A), I get troubles..
P(B/A) should be
$ P(B\mid A) = \frac{P(A\mid B)P(B)}{P(A)} $
$P(B\mid A)$ is $2/3$, $P(A)$ is $1/2$, $P(B)$ I think is $5/7$, but how about $P(A\mid B)$?
Is it meaningful asking for the probability that I choose urn 1 knowing that I extracted a white ball? I think not but I'm unsure... am I asking the probability that the urn was the first known the ball extracted was white?