Alyssa and Gabriel play the following game. Alyssa rolls a die. If she rolls a 1,2 , or 3 , Gabriel gives Alyssa \( \$ 4 \). If Alyssa rolls a 4 or 5 , Gabriel gives Alyssa \( \$ 1 \) However, if Alyss a rolls a 6 , she gives Gabriel \( \$ 17 \). a) Determine Alyssa's expectation. b) Determine Gabriel's expectation. a) Alyssa's expectation is \( \square \) dollars. (Round to the nearest cent.) b) Gabriel's expectation is \( \square \) dollars. (Round to the nearest cent.)

To find Alyssa's and Gabriel's expectations, we first need to analyze the outcomes of the die rolls and their corresponding monetary values. ### Step 1: Calculate Alyssa's Expectation Alyssa's possible outcomes based on her die roll are: - Rolling a 1, 2, or 3: She receives $4. - Rolling a 4 or 5: She receives $1. - Rolling a 6: She loses $17. Next, we calculate the probabilities: - The probability of rolling a 1, 2, or 3 (Alyssa gets $4): \( \frac{3}{6} = \frac{1}{2} \) - The probability of rolling a 4 or 5 (Alyssa gets $1): \( \frac{2}{6} = \frac{1}{3} \) - The probability of rolling a 6 (Alyssa loses $17): \( \frac{1}{6} \) Now we calculate her expectation \( E(Alyssa) \): \[ E(Alyssa) = \left( \frac{1}{2} \times 4 \right) + \left( \frac{1}{3} \times 1 \right) + \left( \frac{1}{6} \times (-17) \right) \] Calculating each term: \[ E(Alyssa) = 2 + \frac{1}{3} - \frac{17}{6} \] To combine these, we find a common denominator (which is 6): \[ E(Alyssa) = \frac{12}{6} + \frac{2}{6} - \frac{17}{6} = \frac{12 + 2 - 17}{6} = \frac{-3}{6} = -0.50 \] Thus, Alyssa's expectation is \(-0.50\) dollars. ### Step 2: Calculate Gabriel's Expectation Gabriel's expectation is the opposite of Alyssa's since any money Alyssa gains is a loss for Gabriel and vice versa. Thus, we can directly use Alyssa's expectation to find Gabriel's: \[ E(Gabriel) = -E(Alyssa) = -(-0.50) = 0.50 \] So, Gabriel's expectation is \(0.50\) dollars. ### Final Answers a) Alyssa's expectation is -0.50 dollars. b) Gabriel's expectation is 0.50 dollars.