1. **State the problem:** You draw one bill from a hat containing bills of values $5$, $10$, $20$, and $100$. Each bill is equally likely to be drawn. We want to find the expected value of the amount you get. 2. **Formula for expected value:** The expected value $E$ of a discrete random variable is given by $$E = \sum (\text{value} \times \text{probability})$$ 3. **Calculate probabilities:** Since there are 4 bills and each is equally likely, $$P(5) = P(10) = P(20) = P(100) = \frac{1}{4}$$ 4. **Calculate expected value:** $$E = 5 \times \frac{1}{4} + 10 \times \frac{1}{4} + 20 \times \frac{1}{4} + 100 \times \frac{1}{4}$$ $$= \frac{5 + 10 + 20 + 100}{4}$$ 5. **Simplify numerator:** $$5 + 10 + 20 + 100 = 135$$ 6. **Final expected value:** $$E = \frac{135}{4} = 33.75$$ **Answer:** The expected value of the game to you is **33.75** dollars.