$A$ bag contains $19$ unbiased coins and one coin with head on both sides. One coin is drawn at random and tossed,and a head turns up. If the probability that the drawn coin was unbiased is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $n^2 - m^2$ is equal to:

Three urns $A$,$B$,and $C$ contain $7$ red,$5$ black; $5$ red,$7$ black; and $6$ red,$6$ black balls,respectively. One of the urns is selected at random and a ball is drawn from it. If the ball drawn is black,then the probability that it is drawn from urn $A$ is:

In a school there are $3$ sections $A, B$ and $C$. Section $A$ contains $20$ girls and $30$ boys,section $B$ contains $40$ girls and $20$ boys and section $C$ contains $10$ girls and $30$ boys. The probabilities of selecting the section $A, B$ and $C$ are $0.2, 0.3$ and $0.5$ respectively. If a student selected at random from the school is a girl,then the probability that she belongs to section $A$ is

$A$ man is known to speak the truth $2$ out of $3$ times. If he throws a die and reports that it is $6$,then the probability that it is actually $5$ is:

$A$ bag contains $6$ balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least $5$ black balls is

$A$ person is known to speak truth in $3$ out of $5$ times. If he throws a die and reports that it is six,then the probability that it is actually six,is