Answer :
To determine if this game is fair, we need to look at the probabilities of each player winning based on the outcomes of tossing two coins.
Let's go through the possible outcomes and their probabilities step-by-step:
1. Possible Outcomes:
- Heads & Heads
- Heads & Tails
- Tails & Tails
- Tails & Heads
Each coin has 2 sides, and there are two coins. Thus, we have a total of [tex]\(2 \times 2 = 4\)[/tex] possible outcomes. The probability of each individual outcome (Heads & Heads, Heads & Tails, etc.) is [tex]\(\frac{1}{4}\)[/tex] because each outcome is equally likely.
2. Winning Conditions:
- You win if both coins land on heads.
- Your friend wins if the coins land on different sides (one heads and one tails).
Now, let's determine the probability for each winning condition:
- Your win condition: Both coins land on heads.
- Probability of Heads & Heads is [tex]\(\frac{1}{4}\)[/tex].
- Friend's win condition: One coin shows heads, and the other shows tails.
- Probability of Heads & Tails: [tex]\(\frac{1}{4}\)[/tex].
- Probability of Tails & Heads: [tex]\(\frac{1}{4}\)[/tex].
Adding these two probabilities together:
[tex]\[ \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \][/tex]
This means your friend has a [tex]\(\frac{1}{2}\)[/tex] probability of winning.
3. Other outcome:
- Both coins land on tails.
- Probability of Tails & Tails is [tex]\(\frac{1}{4}\)[/tex].
4. Game Fairness Determination:
For the game to be fair, the probabilities of you winning and your friend winning should be equal.
- Probability that you win: [tex]\(\frac{1}{4}\)[/tex] (since both coins show heads)
- Probability that your friend wins: [tex]\(\frac{1}{2}\)[/tex] (since one coin shows heads, and the other shows tails)
Since [tex]\(\frac{1}{4} \neq \frac{1}{2}\)[/tex], the probabilities are not equal. Thus, the game is not fair.
### Conclusion:
The game is not fair. The probability of you winning is [tex]\(\frac{1}{4}\)[/tex], whereas the probability of your friend winning is [tex]\(\frac{1}{2}\)[/tex]. Hence, option A is incorrect because it suggests that both you and your friend have the same probability of winning, which is not the case.
Let's go through the possible outcomes and their probabilities step-by-step:
1. Possible Outcomes:
- Heads & Heads
- Heads & Tails
- Tails & Tails
- Tails & Heads
Each coin has 2 sides, and there are two coins. Thus, we have a total of [tex]\(2 \times 2 = 4\)[/tex] possible outcomes. The probability of each individual outcome (Heads & Heads, Heads & Tails, etc.) is [tex]\(\frac{1}{4}\)[/tex] because each outcome is equally likely.
2. Winning Conditions:
- You win if both coins land on heads.
- Your friend wins if the coins land on different sides (one heads and one tails).
Now, let's determine the probability for each winning condition:
- Your win condition: Both coins land on heads.
- Probability of Heads & Heads is [tex]\(\frac{1}{4}\)[/tex].
- Friend's win condition: One coin shows heads, and the other shows tails.
- Probability of Heads & Tails: [tex]\(\frac{1}{4}\)[/tex].
- Probability of Tails & Heads: [tex]\(\frac{1}{4}\)[/tex].
Adding these two probabilities together:
[tex]\[ \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \][/tex]
This means your friend has a [tex]\(\frac{1}{2}\)[/tex] probability of winning.
3. Other outcome:
- Both coins land on tails.
- Probability of Tails & Tails is [tex]\(\frac{1}{4}\)[/tex].
4. Game Fairness Determination:
For the game to be fair, the probabilities of you winning and your friend winning should be equal.
- Probability that you win: [tex]\(\frac{1}{4}\)[/tex] (since both coins show heads)
- Probability that your friend wins: [tex]\(\frac{1}{2}\)[/tex] (since one coin shows heads, and the other shows tails)
Since [tex]\(\frac{1}{4} \neq \frac{1}{2}\)[/tex], the probabilities are not equal. Thus, the game is not fair.
### Conclusion:
The game is not fair. The probability of you winning is [tex]\(\frac{1}{4}\)[/tex], whereas the probability of your friend winning is [tex]\(\frac{1}{2}\)[/tex]. Hence, option A is incorrect because it suggests that both you and your friend have the same probability of winning, which is not the case.
