$A$ bag contains $3$ red, $5$ black and $7$ blue balls. If three balls are drawn at random simultaneously from the bag, then the probability of getting at least two blue balls is

$A$ committee has to be made of $5$ members from $6$ men and $4$ women. The probability that at least one woman is present in the committee is:

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2,$ and $r_3$ are the numbers obtained on the die,what is the probability that $\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0$?

Let $X = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : \frac{x^2}{8} + \frac{y^2}{20} < 1 \text{ and } y^2 < 5x\}$. Three distinct points $P, Q,$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q,$ and $R$ form a triangle whose area is a positive integer is

Let $m, n$ be two distinct integers chosen randomly from the set $\{0, 1, 2, \ldots, 99\}$. Then,the probability that $4^m + 4^n + 3$ is divisible by $5$ lies in the interval

Ten persons with badges numbered $1$ to $10$ are in a room. If three of them are asked to leave the room, then the probability that the person with the smallest badge number among the three who left is $5$, is: