1. **State the problem:** John Underpar buys one ticket in a raffle where 80 tickets are sold at 5 each. The prize is a golf club valued at 300. We analyze his possible monetary outcomes, their probabilities, the probability distribution, expected value, and the club's expected return. 2. **Possible monetary outcomes:** - If John wins, he gains the prize value minus the ticket cost: $$300 - 5 = 295$$. - If John loses, he loses the ticket cost: $$-5$$. 3. **Probabilities of each outcome:** - Probability of winning: $$\frac{1}{80}$$. - Probability of losing: $$\frac{79}{80}$$. 4. **Probability distribution:** | Outcome | Probability | |---------|-------------| | 295 | $$\frac{1}{80}$$ | | -5 | $$\frac{79}{80}$$ | 5. **Expected value (mean) calculation:** Use the formula for expected value: $$E(X) = \sum (x_i \times P(x_i))$$ Calculate: $$E(X) = 295 \times \frac{1}{80} + (-5) \times \frac{79}{80}$$ $$= \frac{295}{80} - \frac{395}{80}$$ $$= \frac{295 - 395}{80} = \frac{-100}{80} = -1.25$$ 6. **Interpretation:** The expected value of -1.25 means on average John loses 1.25 per ticket bought over many trials. 7. **Expected return to the Club:** The club sells 80 tickets at 5 each, total revenue: $$80 \times 5 = 400$$ They pay out the prize valued at 300, so expected return: $$400 - 300 = 100$$ This means the club expects to make 100 from the raffle.