If two smallest squares are chosen at random on a chessboard, then the probability of choosing these squares such that they do not have a side in common is:

Two decks of playing cards are well shuffled and $26$ cards are randomly distributed to a player. Then, the probability that the player gets all distinct cards is

Let $\alpha$ be a root of $x^2+x+1=0$ and suppose that a fair die is thrown $3$ times. If $a, b,$ and $c$ are the numbers shown on the die,then the probability that $\alpha^a+\alpha^b+\alpha^c=0$ is

An unbiased coin is tossed. If the result is a head,a pair of unbiased dice is rolled and the sum of the numbers on the two faces is noted. If the result is a tail,a card from a well-shuffled pack of eleven cards numbered $2, 3, 4, \dots, 12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is:

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ 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: