If $3$ numbers are chosen from the first $15$ natural numbers,what is the probability that the numbers are in arithmetic progression?

$\omega$ is a complex cube root of unity. When an unbiased die is thrown $3$ times, if $\beta_1, \beta_2, \beta_3$ are the numbers appeared on the die, then the probability that $\beta_1, \beta_2$ and $\beta_3$ satisfy $\omega^{\beta_1}+\omega^{\beta_2}=-\omega^{\beta_3}$ is

Two points are randomly chosen on the circumference of a circle of radius $r$. The probability that the distance between the two points is at least $r$ is equal to

Seven balls are drawn simultaneously from a bag containing $5$ white and $6$ green balls. The probability of drawing $3$ white and $4$ green balls is:

$A$ bag contains $4$ red and $4$ blue balls. Four balls are drawn one by one from the bag. Find the probability that the drawn balls are of alternating colors.

$A$ coin is tossed $2n$ times. The chance that the number of times one gets head is not equal to the number of times one gets tail is