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

If $S$ is the sample space of a random experiment $\xi$ and $P$ is a probability function defined on the power set $\mathcal{P}(S)$ of $S$,then which one of the following is not satisfied by $P$?
$(i)$ $P(\phi) = 0$
(ii) If $E^c$ is the complementary event of $E$,then $P(E^c) = 1 - P(E)$
(iii) $0 \leq P(E) \leq 1, \forall E \subseteq S$
(iv) If $E_1 \subseteq E_2$,then $P(E_2) \leq P(E_1)$

$12$ balls are distributed among $3$ boxes. The probability that the first box will contain exactly $3$ balls is:

Let $S = \{1, 2, \dots, 20\}$. $A$ subset $B$ of $S$ is said to be "nice" if the sum of the elements of $B$ is $203$. Then the probability that a randomly chosen subset of $S$ is "nice" is

$A$ box contains $b$ blue balls and $r$ red balls. $A$ ball is drawn randomly from the box and is returned to the box with another ball of the same colour. The probability that the second ball drawn from the box is blue,is

In a single throw of three dice,the probability of getting a sum at least $5$ is