In an entrance test, there are multiple-choice questions. There are four possible answers to each question, of which one is correct. The probability that a student knows the answer to a question is $9/10$. If he gets the correct answer to a question, then the probability that he was guessing is

$A$ 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:

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 $\frac{4}{13}$

$A$ man is known to speak the truth $7$ out of $10$ times. After throwing a die with $100$ faces marked $1, 2, 3, \dots, 100$,the man reports that he got a prime number. What is the probability that it is actually a prime number?

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?

$A$ bag $P$ contains $3$ blue and $5$ red balls. Another bag $Q$ contains $4$ blue and $6$ red balls. $A$ ball is drawn at random from one of the bags and is found to be red. The probability that it is from bag $Q$ is