$A$ bag contains $2n$ coins,out of which $n-1$ are unfair with heads on both sides and the remaining are fair. One coin is picked from the bag at random and tossed. If the probability that a head appears in the toss is $\frac{41}{56}$,then the number of unfair coins in the bag is:

There are four machines and it is known that exactly two of them are faulty. They are tested one by one,in a random order,until both the faulty machines are identified. The probability that only two tests are needed is:

An ordinary cube has four blank faces,one face marked $2$,and another marked $3$. The probability of obtaining a total of exactly $12$ in $5$ throws is:

If the letters of the word $REGULATIONS$ are arranged in such a way that the relative positions of the letters of the word $GULATIONS$ remain the same,then the probability that there are exactly $4$ letters between $R$ and $E$ is

An urn contains marbles of four colours: red,white,blue,and green. When four marbles are drawn without replacement,the following events are equally likely:
$1.$ The selection of four red marbles.
$2.$ The selection of one white and three red marbles.
$3.$ The selection of one white,one blue,and two red marbles.
$4.$ The selection of one marble of each colour.
The smallest total number of marbles satisfying the given condition is:

$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$. Then the probabilities of the two events are respectively: