If three dice are thrown at a time,then the probability of getting the sum of the numbers on them as a prime number is

Let $X$ and $Y$ be events such that $P(X \cup Y) = P(X \cap Y).$
Statement-$1$: $P(X \cap Y) = P(X' \cap Y') = 0$
Statement-$2$: $P(X) + P(Y) = 2P(X \cap Y).$

In a battery manufacturing factory,machines $P$,$Q$ and $R$ manufacture $20 \%$,$30 \%$ and $50 \%$ respectively of the total output. The chances that a defective battery is produced by these machines are $1 \%$,$1.5 \%$ and $2 \%$ respectively. If a battery is selected at random from the production,then the probability that it is defective is

$A$ bag contains $20$ coins. If the probability that the bag contains exactly $4$ biased coins is $1/3$ and the probability that it contains exactly $5$ biased coins is $2/3$,then the probability that all the biased coins are sorted out from the bag in exactly $10$ draws is:

Let $B_{i} (i=1, 2, 3)$ be three independent events in a sample space. The probability that only $B_{1}$ occurs is $\alpha$,only $B_{2}$ occurs is $\beta$,and only $B_{3}$ occurs is $\gamma$. Let $p$ be the probability that none of the events $B_{i}$ occurs,and these $4$ probabilities satisfy the equations $(\alpha - 2\beta)p = \alpha\beta$ and $(\beta - 3\gamma)p = 2\beta\gamma$ (All the probabilities are assumed to lie in the interval $(0, 1)$). Then $\frac{P(B_{1})}{P(B_{3})}$ is equal to ..........

If $A$ and $B$ are two events,then the probability of the event that at most one of $A$ and $B$ occurs is: