A game involves tossing two coins. A player wins \( \$ 1.00 \) if both tosses result in heads. What should you pay to play this game in order to make fair game? Explain your reasoning. You should pay \( \$ \square \) to play this game in order to make it a fair game.

To determine a fair price to play the game, we need to calculate the **expected value** of the game. A fair game means that the cost to play equals the expected winnings, ensuring neither the player nor the organizer has an advantage over time. **Here's how to calculate it:** 1. **Determine the possible outcomes when tossing two coins:** - **HH (Heads-Heads):** Player wins \$1.00 - **HT (Heads-Tails):** Player wins \$0 - **TH (Tails-Heads):** Player wins \$0 - **TT (Tails-Tails):** Player wins \$0 2. **Calculate the probability of each outcome:** - There are 4 possible outcomes, each equally likely. - **Probability of HH:** \( \frac{1}{4} \) - **Probability of HT:** \( \frac{1}{4} \) - **Probability of TH:** \( \frac{1}{4} \) - **Probability of TT:** \( \frac{1}{4} \) 3. **Determine the expected winnings:** \[ \text{Expected Winnings} = \left(\frac{1}{4} \times \$1.00\right) + \left(\frac{3}{4} \times \$0\right) = \$0.25 \] 4. **Set the cost equal to the expected winnings for a fair game:** \[ \text{Fair Price to Play} = \$0.25 \] **Conclusion:** You should pay **\$0.25** to play this game in order to make it a fair game.