There are four fair dice $D_1, D_2, D_3$ and $D_4$. Each has six faces numbered $1, 2, 3, 4, 5$ and $6$. They are rolled one by one. What is the probability that the number shown on $D_4$ is equal to at least one of the numbers shown on $D_1, D_2$ and $D_3$ (in $/216$)?

Two numbers are selected randomly from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ without replacement one by one. The probability that the minimum of the two numbers is divisible by $3$ or the maximum of the two numbers is divisible by $4$ is:

There are four machines and it is known that exactly two of them are faulty. They are tested one by one,in a random order,until both the faulty machines are identified. The probability that only two tests are needed is:

Consider the set $A_n$ of points $(x, y)$ such that $0 \leq x \leq n, 0 \leq y \leq n$,where $n, x, y$ are integers. Let $S_n$ be the set of all lines passing through at least two distinct points from $A_n$. Suppose we choose a line $l$ at random from $S_n$. Let $P_n$ be the probability that $l$ is tangent to the circle $x^2+y^2=n^2\left(1+\left(1-\frac{1}{\sqrt{n}}\right)^2\right)$. Then,the limit $\lim _{n \rightarrow \infty} P_n$ is

For the three events $A, B$ and $C$,$P$ (exactly one of the events $A$ or $B$ occurs) = $P$ (exactly one of the events $B$ or $C$ occurs) = $P$ (exactly one of the events $C$ or $A$ occurs) = $p$ and $P$ (all the three events occur simultaneously) = $p^2$,where $0 < p < 1/2$. Then the probability of at least one of the three events $A, B$ and $C$ occurring is

For two events $A$ and $B$,if $P(A) = P(A|B) = \frac{1}{4}$ and $P(B|A) = \frac{1}{2}$,then which of the following is true?