If $E_1, E_2, \ldots, E_n$ are independent events such that $P(E_r) = \frac{1}{1+r}$ for $r = 1, 2, \ldots, n$, then the probability that at least one of $E_1, E_2, \ldots, E_n$ happens is

$A$ random variable $X$ has the following probability distribution:
| $X=x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $P(X=x)$ | $0.15$ | $0.23$ | $0.12$ | $0.20$ | $0.08$ | $0.10$ | $0.05$ | $0.07$ |
For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 5\}$,find $P(E \cup F)$.

Let $X_n = \{1, 2, 3, \ldots, n\}$ and let a subset $A$ of $X_n$ be chosen such that every pair of elements of $A$ differ by at least $3$. (For example,if $n = 5$,$A$ can be $\phi, \{2\}$ or $\{1, 5\}$ among others). When $n = 10$,let the probability that $1 \in A$ be $p$ and let the probability that $2 \in A$ be $q$. Then,

Consider the system of equations $ax+by=0, cx+dy=0$,where $a, b, c, d \in \{0, 1\}$.
$STATEMENT-1$: The probability that the system of equations has a unique solution is $3/8$.
$STATEMENT-2$: The probability that the system of equations has a solution is $1$.

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

$A$ bag contains $19$ red balls and $19$ black balls. Two balls are chosen at a time repeatedly and discarded if they are of the same colour,but if they are different,the black ball is discarded and the red ball is returned to the bag. The probability that this process will terminate with one red ball is