$A$ boy speaks the truth in $3$ out of $5$ times. If he throws a die and reports that the number appeared on it is $5$,then the probability that it is actually $5$ is:

Let $U_1$ and $U_2$ be two urns such that $U_1$ contains $3$ white and $2$ red balls,and $U_2$ contains only $1$ white ball. $A$ fair coin is tossed. If it shows heads,$1$ ball is drawn at random from $U_1$ and transferred to $U_2$. If it shows tails,$2$ balls are drawn at random from $U_1$ and transferred to $U_2$. Now,$1$ ball is drawn at random from $U_2$. Given that the ball drawn from $U_2$ is white,what is the probability that the coin showed heads (in $/23$)?

Bag $A$ contains $2$ white and $3$ red balls and bag $B$ contains $4$ white and $5$ red balls. If one ball is drawn at random from one of the bags and is found to be red,then the probability that it was drawn from the bag $B$ is

Bag $A$ contains $2$ white and $3$ red balls,and bag $B$ contains $4$ white and $5$ red balls. One ball is drawn at random from one of the two bags and it is found to be red. Find the probability that the ball was drawn from bag $B$.

There are two coins,one unbiased with probability $\frac{1}{2}$ of getting heads and the other one is biased with probability $\frac{3}{4}$ of getting heads. $A$ coin is selected at random and tossed. It shows heads up. Then,the probability that the unbiased coin was selected is

Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. Then one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$,the probability that it is found to be red,is