Fifteen football players of a club-team are given $15$ $T$-shirts with their names written on the backside. If the players pick up the $T$-shirts randomly,then the probability that at least $3$ players pick the correct $T$-shirt is

$A$ and $B$ throw a die alternatively till one of them gets a '$6$' and wins the game. Find their respective probabilities of winning,if $A$ starts first.

Let the probability of getting a head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64x^2 + 5Nx + 1 = 0$ has no real root is $\frac{p}{q}$,where $p$ and $q$ are co-prime,then $q - p$ is equal to

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:

If the integers $m$ and $n$ are chosen at random between $1$ and $100$,then the probability that a number of the form $7^m + 7^n$ is divisible by $5$ equals

$U_1, U_2, U_3$ are three urns. $U_1$ contains $5$ red,$3$ white,$2$ black balls; $U_2$ contains $4$ red,$4$ white,$2$ black balls and $U_3$ contains $3$ red,$4$ white,$3$ black balls. If a ball is chosen at random from an urn chosen at random,then the probability of not getting a black ball is