1. **State the problem:** We need to find the expected value of the winnings from a game with given payout probabilities. 2. **Formula:** The expected value $E(X)$ of a discrete random variable $X$ is given by: $$E(X) = \sum (x_i \times p_i)$$ where $x_i$ are the payouts and $p_i$ are their corresponding probabilities. 3. **Apply the formula:** $$E(X) = 3 \times 0.01 + 4 \times 0.04 + 5 \times 0.10 + 6 \times 0.20 + 7 \times 0.65$$ 4. **Calculate each term:** $$3 \times 0.01 = 0.03$$ $$4 \times 0.04 = 0.16$$ $$5 \times 0.10 = 0.50$$ $$6 \times 0.20 = 1.20$$ $$7 \times 0.65 = 4.55$$ 5. **Sum the terms:** $$E(X) = 0.03 + 0.16 + 0.50 + 1.20 + 4.55 = 6.44$$ 6. **Final answer:** The expected value of the winnings is **6.44** (rounded to the nearest hundredth).