Total Probability Theorem

A bag contains $20$ coins. If the probability that the bag contains exactly $4$ biased coin is $\frac{1}{3}$ and that of exactly $5$ biased coin is $\frac{2}{3}$, then the probability that all the biased coin are sorted out from the bag in exactly $10$ draws is

There are two urns. There are $m$ white & $n$ black balls in the first urn and $p$ white & $q$ black balls in the second urn. One ball is taken from the first urn & placed into the second. Now, the probability of drawing a white ball from the second urn is

A purse contains $2$ six-sided dice. One is a normal fair die, while the other has two $1^{\text{'}}\text{s},$ two $3^{\text{'}}\text{s},$ and two $5^{\text{'}}\text{s}$. A die is picked up and rolled. Because of some secret magnetic attraction of the unfair die, there is $75\%$ chance of picking the unfair die and a $25\%$ chance of picking a fair die. The die is rolled and shows up the face $3$. The probability that a fair die was picked up is