A box contains $5$ radio tubes of which $2$ are defective. The tubes are tested one after the other until the $2$ defective tubes are discovered. The probability that the process stopped on the

(a) second test is $\frac{1}{a},$ then $a$ is

(b) third test is $\frac{115}{b},$ then $b$ is

(c) If the process stopped on the third test, find the probability that the first tube is non-defective is $\frac{x}{y}$, then $\left(x+y\right)$ is

Two distinct numbers $a$ and $b$ are chosen randomly from the set $\left\{2,2^2,2^{3,}{2}^4,\mathrm{.....},2^{25}\right\}$. Find the probability that ${\log }_{a}b$ is an integer.

A uniform unbiased die is constructed in shape of a regular tetrahedron with faces numbered $2,2,3$ and $4$ and the score is taken from the face on which the die lands. If two such dice are thrown together, find the probability of scoring.

(a) exactly $6$ on each $3$ successive throws.

(b) more than $4$ on atleast one of the three successive throws.

A player tosses an unbiased coin and is to score two points for every head turned up and one point for every tail turned up. If $P_n$ denotes the probability that his score is exactly $n$ points, prove that

$P_n-P_{n-1}=\frac{1}{2}\left(P_{n-2}-P_{n-1}\right),n\ge 3$