To determine if this game is fair, we need to analyze the probabilities of you and your friend winning the game based on the given possible outcomes.
### Step-by-Step Solution:
1. List all possible outcomes:
- The game involves tossing two coins. The possible outcomes are:
1. Heads (H) and Heads (H)
2. Heads (H) and Tails (T)
3. Tails (T) and Tails (T)
4. Tails (T) and Heads (H)
2. Identify the outcomes in which you win:
- You win if both coins show heads or both show tails.
- The winning outcomes for you are:
1. (Heads, Heads)
2. (Tails, Tails)
3. Count the number of winning outcomes for you:
- There are 2 outcomes where you win: (H, H) and (T, T).
4. Identify the outcomes in which your friend wins:
- Your friend wins if one coin shows heads and the other shows tails.
- The winning outcomes for your friend are:
1. (Heads, Tails)
2. (Tails, Heads)
5. Count the number of winning outcomes for your friend:
- There are 2 outcomes where your friend wins: (H, T) and (T, H).
6. Calculate the total number of possible outcomes:
- The total number of possible outcomes when tossing two coins is 4.
7. Calculate the probability of you winning:
- Probability (You win) = Number of outcomes where you win / Total number of outcomes
- Probability (You win) = 2 / 4 = 0.5
8. Calculate the probability of your friend winning:
- Probability (Your friend wins) = Number of outcomes where your friend wins / Total number of outcomes
- Probability (Your friend wins) = 2 / 4 = 0.5
9. Determine if the game is fair:
- A game is considered fair if all players have equal chances of winning.
- In this case, both you and your friend have a probability of 0.5 (or [tex]\(\frac{1}{2}\)[/tex]) of winning.
### Conclusion:
Since both you and your friend have an equal probability of 0.5 (or [tex]\(\frac{1}{2}\)[/tex]) of winning, the game is fair. Therefore, the correct answer is:
B. Yes. You and your friend each have a [tex]\(\frac{1}{2}\)[/tex] probability of winning.
