The probability that a year chosen at random from the $22^{nd}$ century will have $53$ Sundays is

The probabilities that players $A$ and $B$ of a team are selected for the captaincy for a tournament are $0.6$ and $0.4$,respectively. If $A$ is selected as the captain,the probability that the team wins the tournament is $0.8$ and if $B$ is selected as the captain,the probability that the team wins the tournament is $0.7$. Then the probability,that the team wins the tournament,is:

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?

Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happen is $\frac{1}{12}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2},$ then

Let $A, B,$ and $C$ be $3$ independent events such that $P(A) = 1/3, P(B) = 1/2,$ and $P(C) = 1/4$. Find the probability that exactly $2$ of the $3$ events occur.

$A$ die is rolled three times. The probability of getting their sum equal to a prime number of the form $4n+1$ is