A set of 20 cards consists of 12 red cards and 8 black cards. The cards are shuffled thoroughly and you choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and you again choose a card at random, observe its color, and replace it in the set. This is done a total of six times. Let X be the number of red cards observed in these six trials. The variance of X is

I know the correct answer! Can anyone show me how to do these types of problems?

This situiation yields the binomial probability distribution because each trial has only two possible outcomes - in this case red or black. Let p = probability of choosing red for any one trial, and q = probability for choosing black (note that q = 1-p). If you do N trials the expected number of occurrences of red is E[red] = Np, and the variance is Var[red] = npq.