A list of all possible values of the discrete random variable X, together with their associated probabilities, is called a probability distribution. It is often helpful to show the probability distribution in a table.

Stephanie is very forgetful. Every time she logs in to her online bank she only has a 40% chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

(a) Draw a fully labelled tree diagram to illustrate this situation.

(b) Let X be the number of unsuccessful attempts Stephanie makes on any day that she tries to log in to her bank. Construct a table to show the probability distribution of X.

(c) Calculate the expected number of unsuccessful attempts made by Stephanie on any day that she tries to log in.

(d) Calculate variance.of X.