$A$ box contains $n$ coins,$m$ of which are fair and the rest are biased. When a biased coin is tossed,the probability of getting a head is twice as likely as tail. $A$ coin is drawn from the box at random and is tossed twice. It is found that the first time it shows head and the second time it shows tail. Then,the probability that the coin drawn is fair is

$A$ person goes to office by car, scooter, bus, and train, the probabilities of which are $1/7, 3/7, 2/7,$ and $1/7$ respectively. The probability that he reaches office late if he takes a car, scooter, bus, or train is $2/9, 1/9, 4/9,$ and $1/9$ respectively. Given that he reached the office in time, the probability that he travelled by car is: (in $/7$)

There are three bags $B_1, B_2$ and $B_3$. The bag $B_1$ contains $5$ red and $5$ green balls,$B_2$ contains $3$ red and $5$ green balls,and $B_3$ contains $5$ red and $3$ green balls. Bags $B_1, B_2$ and $B_3$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. $A$ bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
$(1)$ Probability that the selected bag is $B_3$ and the chosen ball is green equals $\frac{3}{20}$
$(2)$ Probability that the chosen ball is green equals $\frac{39}{80}$
$(3)$ Probability that the chosen ball is green,given that the selected bag is $B_3$,equals $\frac{3}{8}$
$(4)$ Probability that the selected bag is $B_3$,given that the chosen ball is green,equals $\frac{4}{13}$

$A$ 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:

Bag $A$ contains $3$ white and $5$ black balls, while bag $B$ contains $4$ white and $3$ black balls. $A$ ball is selected at random from bag $A$ and put into bag $B$. If a ball is now selected at random from bag $B$, what is the probability that this ball is white?

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