Find the variance of the distribution:

Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.

The random variable X can take only the values 0, 1, 2. Given that P (X = 0) = P (X = 1) = p and that E(X²) = E[X], find the value of p.

A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. It A starts the game, find the probability of winning the game by A in third throw of the pair of dice.

Two dice are tossed. Find whether the following two events A and B are independent:

A = {(x, y): x+y = 11} and B = {(x, y): x ≠ 5}

where (x, y) denotes a typical sample point.