If three boxes contain $3$ white and $1$ black,$2$ white and $2$ black,and $1$ white and $3$ black balls respectively,and one ball is chosen at random from each box,what is the probability of selecting $2$ white and $1$ black ball?

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

Let there be three independent events $E_{1}, E_{2}$ and $E_{3}$. The probability that only $E_{1}$ occurs is $\alpha$,only $E_{2}$ occurs is $\beta$ and only $E_{3}$ occurs is $\gamma$. Let $p$ denote the probability that none of the events occur,which satisfies the equations $(\alpha - 2\beta)p = \alpha\beta$ and $(\beta - 3\gamma)p = 2\beta\gamma$. All the given probabilities are assumed to lie in the interval $(0, 1)$. Then,$\frac{\text{Probability of occurrence of } E_{1}}{\text{Probability of occurrence of } E_{3}}$ is equal to ..........

Cards are drawn one after the other without replacement from a well-shuffled pack of cards until an ace card appears. If the probability that exactly $5$ cards are drawn before the first ace card appears is $\frac{4}{49}\left(\frac{p_1 \cdot p_2 \cdot p_3}{p_4 \cdot p_5 \cdot p_6}\right)$, where $p_i$ is prime for $i=1, 2, 3, 4, 5, 6$, then $(\max \{p_i\} - \min \{p_i\}) = $

If $7$ dice are thrown simultaneously,then the probability that all six digits appear on the upper faces is equal to -

Three screws are drawn at random from a lot of $50$ screws containing $5$ defective ones. The probability of the event that all $3$ screws drawn are non-defective,assuming that the drawing is $(a)$ with replacement and $(b)$ without replacement,is respectively: