The probability of getting a number greater than $2$ in throwing a die is

Let $S$ be the sample space of the random experiment of throwing simultaneously two unbiased dice and $E_k = \{(a, b) \in S : ab = k\}$. If $p_k = P(E_k)$, then which of the following is correct?

$X$ speaks truth in $60\%$ and $Y$ in $50\%$ of the cases. The probability that they contradict each other in narrating the same incident is

Suppose $3$ bulbs are selected at random from a lot. Each bulb is tested and classified as defective $(D)$ or non-defective $(N)$. Write the sample space of this experiment.

$A$ mathematics problem is given to three students $A, B$,and $C$. The probabilities of them solving the problem are $1/2, 1/3$,and $1/4$ respectively. What is the probability that the problem is solved?

$A$ man and a woman appear in an interview for two vacancies in the same post. The probability of the man's selection is $1/4$ and that of the woman's selection is $1/3$. What is the probability that none of them will be selected?