$A$ man is known to speak the truth $4$ out of $5$ times. He throws a die and reports that it is a six. The probability that it was actually a six is

Two coins are available,one fair and the other two-headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probability $\frac{3}{4}$. Given that the outcome is head,the probability that the two-headed coin was chosen,is

$A$ bag contains $(N+1)$ coins: $N$ fair coins,and one coin with 'Head' on both sides. $A$ coin is selected at random and tossed. If the probability of getting 'Head' is $\frac{9}{16}$,then $N$ is equal to:

$A$ signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively,is received by station $A$ and then transmitted to station $B$. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station $B$ is green,then the probability that the original signal was green is

In a toy factory,machines $A, B$,and $C$ are used to manufacture $30 \%, 40 \%$,and $30 \%$ of the output,respectively. The probabilities of toys made by machines $A, B$,and $C$ being defective are $2 \%, 3 \%$,and $1 \%$,respectively. $A$ toy is taken from the factory and is found to be defective. The probability that it was manufactured by machine $B$ is

$A$ bag $P$ contains $5$ white marbles and $3$ black marbles. Four marbles are drawn at random from $P$ and are put in an empty bag $Q$. If a marble drawn at random from $Q$ is found to be black,then the probability that all the three black marbles in $P$ were transferred to the bag $Q$ is: