$3$ numbers are chosen from first $15$ natural numbers, then probability that the numbers are in arithmetic progression

Two numbers $a$ and $b$ are chosen at random from the set of first $30$ natural numbers. The probability that ${a^2} - {b^2}$ is divisible by $3$ is

Six points are there on a circle . Two triangles are drawn with no vertex common. What is the probability that none of the sides of the triangles intersect

In a lottery, a person choses six different natural numbers at random from $1$ to $20$ , and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [ Hint order of the numbers is not important.]

Two cards are drawn at random from a pack of $52$ cards. The probability that both are the cards of spade is

An ordinary cube has four blank faces, one face marked $2$ another marked $3$. Then the probability of obtaining a total of exactly $12$ in $5$ throws, is