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

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three persons apply for the same house is

$A$ problem in Algebra is given to two students $A$ and $B$ whose chances of solving it are $\frac{2}{5}$ and $\frac{3}{4}$ respectively. The probability that the problem is solved if both of them try independently is

Two persons $A$ and $B$ throw three unbiased dice one after the other. If $A$ gets a sum of $13$,then the probability that $B$ gets a higher sum is:

$A$ die has two faces each with number $1$,three faces each with number $2$,and one face with number $3$. If the die is rolled once,determine $P(1 \text{ or } 3)$.

Let $A$ and $B$ be two events such that $P(A \cap B) = \frac{1}{6}$, $P(A \cup B) = \frac{31}{45}$, and $P(\bar{B}) = \frac{7}{10}$, then: