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 it a 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 for playing the game, we need to calculate the **expected value** of the game. A fair game is one where the expected value is zero, meaning the cost to play equals the expected winnings. Here's the step-by-step reasoning: 1. **Determine the Probability of Winning:** - Since the game involves tossing two fair coins, each coin has two possible outcomes: Heads (H) or Tails (T). - The possible outcomes for two coins are: HH, HT, TH, TT. - Only one of these four outcomes results in both coins being heads (HH). - **Probability of winning (both heads):** \( \frac{1}{4} \) or 25%. 2. **Calculate the Expected Winnings:** - If you win, you receive \$1.00. - **Expected Winnings:** Probability of winning × Amount won = \( \frac{1}{4} \times \$1.00 = \$0.25 \). 3. **Determine the Fair Price to Play:** - To make the game fair, the cost to play should equal the expected winnings. - Therefore, you should pay **\$0.25** to play the game. **Conclusion:** You should pay **\$0.25** to play this game in order to make it a fair game. $\boxed{0.25}$