Urn A has 4 red, 3 yellow, and 7 blue balls. Urn B has 3 red, 4 yellow, and 12 blue balls. A ball is first selected at random from urn A and put into urn B. Then a ball is selected at random from urn B. If the ball selected from urn B is blue, what is the probability that the ball transfered from urn A to urn B is red?
If the second ball is blue, it is either originally from urn B, or originally from urn A.
The probability for drawing a blue ball that is from the 12 originally among urn B, when drawing from the $19$ balls in urn B after the transfer is: ___
The probability for first transfering one from the 7 blue ball from urn A (among the 14 balls there) and then drawing it again is: ___
So the probability for drawing a blue balls is: ___. Call this $\mathsf P(Y)$
The probability for transferring one from the 4 red balls in urn A and then drawing one from the 12 blue balls in urn B is: ___. Call this $\mathsf P(X\cap Y)$.
You seek $\mathsf P(X\mid Y)$, the probability for transferring a red ball when given that a blue ball will be drawn from the second urn. Use the definition of conditional probability.