Two coins are tossed. You win \( \$ 6.00 \) if both coins are heads and \( \$ 10.00 \) if both coins are tails. You lose \( \$ 8.00 \) if the coins do not match. What is the expected value of this game? Is this a fair game? The expected value is \( \$ \square \).

To determine the expected value of the game and whether it's fair, let's analyze all possible outcomes when two fair coins are tossed: 1. **Both Heads (HH)**: - **Probability**: \(\frac{1}{4}\) - **Outcome**: Win \$6.00 2. **Both Tails (TT)**: - **Probability**: \(\frac{1}{4}\) - **Outcome**: Win \$10.00 3. **One Head and One Tail (HT or TH)**: - **Probability**: \(\frac{1}{2}\) - **Outcome**: Lose \$8.00 **Calculating the Expected Value (E):** \[ E = \left(\frac{1}{4} \times 6\right) + \left(\frac{1}{4} \times 10\right) + \left(\frac{1}{2} \times (-8)\right) \] \[ E = 1.5 + 2.5 - 4 = 0 \] **Interpretation:** - **Expected Value**: \$0.00 - **Fairness**: Since the expected value is zero, the game is **fair**. This means that, on average, you neither gain nor lose money over many repetitions of the game. **Final Answer:** The expected value is \$0.00, and thus the game is fair.