Jar A has 5 yellow and 8 green while jar B has 2 yellow and 6 green marbles. Two marbles are transferred from jar A to jar B and then one marble is taken from jar B.
a) Find the probability that marble drawn from jar B is yellow.
b) Given that the marble drawn is yellow, find the probability that is was originally from jar A.
I got part a correct, answer is $\frac{18}{65}$. However, my part b is wrong.
What I did for part b:
P(yellow from A | yellow) = $\frac{\text{P(yellow and it is from A)}}{\text{P(yellow)}} $
Numerator: P(yellow and it is from A) = $\frac{5}{13}$ $\times$ $\frac{4}{12}$ $\times$ $\frac{2}{10}$ + $\frac{5}{13}$ $\times$ $\frac{8}{12}$ $\times$ $\frac{1}{10}$ $\times$ 2
Denominator: P(yellow) = $\frac{18}{65}$
And I got $\frac{5}{18}$.
But the correct answer is $\frac{55}{234}$. May I ask what did I do wrong?