Question

A lottery has a grand prize of $140,000, five runner-up prizes of $17,500 each, nine third-place prizes of $3500 each, and twenty-three consolation prizes of $280 each. If 420,000 tickets are sold for $1 each and the probability of any one ticket winning is the same as that of any other ticket winning, find the expected return on a $1 ticket. (Round your answer to two decimal places.)

Answer 1

Suppose, random variable X denotes return (in dollars) on a $1 ticket.

Probability mass function of X is as follows.

$\tiny P\left ( X=x \right )=\begin{cases} \frac{1}{420000} & \text{ if } x=140000 \\ \\ \frac{5}{420000} & \text{ if } x=17500 \\ \\ \frac{9}{420000} & \text{ if } x=3500 \\ \\ \frac{23}{420000} & \text{ if } x=280 \\ \\ \frac{419962}{420000} & \text{ if } x=0 \end{cases}$

$\tiny E\left ( X=x \right )= \frac{140000*1}{420000}+ \frac{17500*5}{420000}+ \frac{3500*9}{420000}+ \frac{280*23}{420000}+ \frac{0*419962}{420000}=0.632$

Hence, expected return is $ 0.63 on a $1 ticket.