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A car dealer believes that for a given customer the probability of making a sale is . Define the random variable:

$X = \begin{cases} 0 & \text{if no sale is made} \\ 1 & \text{if a sale is made} \end{cases}$

Answer the following, rounding your answers to four decimal places unless stated otherwise.

(a) What is the mean of $X$ ?

(b) What is the variance of $X$ ? Provide your answer to four decimal places.

Suppose now that the car dealer has customers and believes that, for each, the probability of making a sale is and that customer's individual decision about buying is independent of other's choices.

(c) What is the probability that three sales are made?

(d) What is the expected number of sales?

You can earn partial credit on this problem.