If a third ball is drawn from urn $U_2$ then what is the probability that it is white?

An urn $U_1$ has $3$ white balls and $5$ black balls. $4$ balls were drawn from this urn and put into another empty urn $U_2$. Now $2$ balls were drawn from the urn $U_2$ and were found to be white. If a third ball is drawn from urn $U_2$ then what is the probability that it is white?

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Urn 2 is irrelevant. We can imagine we are looking at the balls as they come out of Urn 1. If the first two are white, there is one white left, so the probability the third is white is $1/6$.
After moving $4$ balls to urn $2$, the probability distribution for the number of white balls is $\{ 0 : \frac{5}{70}, 1: \frac{30}{70}, 2: \frac{30}{70}, 3: \frac{5}{70}\}$. (For example, we can choose 2 white balls in $\binom32 \binom52 = 30$ ways.)
So the chances of pulling $2$ white balls out of urn $2$ are $$\frac{5}{70}\frac34\frac23 + \frac{30}{70}\frac12\frac13= \frac5{140}+\frac{10}{140} = \frac{30}{280}$$ And the chances of pulling $2$ white balls out of urn $2$ and then pulling a third white ball out are $$\frac{5}{70}\frac34\frac23\frac12 = \frac5{280}$$ So the probability of that second event given that the first event happened is $$\frac{5}{30} = \frac16$$