If $A$ and $B$ are two independent events such that $P(\bar{A})=0.75$,$P(A \cup B)=0.65$ and $P(B)=x$,then find the value of $x$.

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,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

$A$ and $B$ are two candidates seeking admission in a college. The probability that $A$ is selected is $0.7$ and the probability that exactly one of them is selected is $0.6$. Find the probability that $B$ is selected.

For two independent events $A$ and $B$,which of the following is true?

The probability that $A$ speaks truth is $75 \%$ and the probability that $B$ speaks truth is $80 \%$. The probability that they contradict each other when asked to speak on a fact is

If three numbers are randomly selected from the set $\{1, 2, 3, \ldots, 50\}$, then the probability that they are in arithmetic progression is