If $10$ different balls are to be placed in $4$ distinct boxes at random,then the probability that two of these boxes contain exactly $2$ and $3$ balls is

If $12$ identical balls are to be placed randomly in $3$ identical boxes,then the probability that one of the boxes contains exactly $3$ balls is

Two coins are tossed. Let $A$ be the event that the first coin shows a head and $B$ be the event that the second coin shows a tail. The two events $A$ and $B$ are:

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B)=\frac{1}{6}$ and $P(\bar{A} \cap \bar{B})=\frac{1}{3}$, then $P(A)$ is equal to (Here, $\bar{E}$ is the complement of the event $E$)

If a year is selected at random from the $22^{nd}$ century,what is the probability that it contains $53$ Sundays (in $/28$)?

In a tournament,there are $12$ players $P_1, P_2, P_3, \dots, P_{12}$ divided into $6$ pairs at random. From each game,a winner is decided based on the game played between the two players of the pair. Assuming each player is of equal strength,what is the probability that exactly one out of $P_1$ and $P_2$ is among the losers?