Two urns contain, respectively, $20$ white and $10$ black balls and $10$ white and $50$ black balls. $5$ balls are transferred from the first urn to the second urn, and then $1$ ball is drawn from the second urn and is found to be white. What is the probability that this white ball comes form first urn?
I can find the probability of 'the ball drawn from the 2nd urn is found to be white', I also can find the probability of 'the five balls transferred are all white when it is known that we found the ball from the 2nd urn is white'... but this particular Question is very confusing
How do I find the probability of drawing a white ball from the second urn?
$W =$ White, $B= $ Black
$X \equiv$ Event , founding a white ball from 2nd urn.
Following events are considered when transferring the balls from 1st urn to 2nd urn :
$A \equiv $ Event, $(1W,4B)$
$B \equiv $ Event, $(2W,3B)$
$C \equiv $ Event, $(3W,2B)$
$D \equiv $ Event, $(4W,1B)$
$E \equiv $ Event, $(5W,0B)$
$P(X) = P(X|A)P(A)+ P(X|B)P(B)+P(X|C)P(C)+P(X|D)P(D)+P(X|E)P(E)$