$A$ bag contains $4$ red and $5$ black balls. Another bag contains $3$ red and $6$ black balls. If one ball is drawn from the first bag and two balls are drawn from the second bag at random,the probability that out of the three balls,two are black and one is red,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?

Three students $S_1, S_2$ and $S_3$ are given a problem to solve. Consider the following events:
$U:$ At least one of $S_1, S_2$ and $S_3$ can solve the problem,
$V: S_1$ can solve the problem,given that neither $S_2$ nor $S_3$ can solve the problem,
$W: S_2$ can solve the problem and $S_3$ cannot solve the problem,
$T: S_3$ can solve the problem.
For any event $E$,let $P(E)$ denote the probability of $E$.
If $P(U)=\frac{1}{2}, P(V)=\frac{1}{10}$ and $P(W)=\frac{1}{12}$,then $P(T)$ is equal to

If $l, m$ represent any two elements (identical or different) of the set $\{1, 2, 3, 4, 5, 6, 7\}$,then the probability that $lx^2 + mx + 1 > 0$ for all $x \in R$ is

If a year is chosen at random from the years having $53$ Sundays and $53$ Mondays,what is the probability that it is a leap year?

Two balls are drawn from an urn containing $7$ white,$6$ red,and $8$ black balls one after the other without replacement. The probability that at least one of them is white is: