To determine whether this is a fair game, we need to evaluate the probabilities of each player winning and then compare these probabilities.
First, let's list all possible outcomes when tossing two coins:
1. Heads, Heads (HH)
2. Heads, Tails (HT)
3. Tails, Heads (TH)
4. Tails, Tails (TT)
Each of these outcomes has an equal probability of occurring because there are two coins and each coin has an equal chance of landing on Heads or Tails. Therefore, the probability of each outcome is:
[tex]\[
\text{Probability of each outcome} = \frac{1}{4}
\][/tex]
Next, let's determine the probability that you win. You win if and only if both coins land on Heads (HH):
- Probability of Heads, Heads (HH) = [tex]\(\frac{1}{4}\)[/tex]
Thus, your probability of winning is:
[tex]\[
\text{Your probability of winning} = \frac{1}{4}
\][/tex]
Let's now determine the probability that your friend wins. Your friend wins if and only if both coins land on Tails (TT):
- Probability of Tails, Tails (TT) = [tex]\(\frac{1}{4}\)[/tex]
Thus, your friend's probability of winning is:
[tex]\[
\text{Friend's probability of winning} = \frac{1}{4}
\][/tex]
Since both you and your friend have an equal probability of winning (both [tex]\(\frac{1}{4}\)[/tex]), the game is fair.
Therefore, the correct answer is:
A. Yes. You and your friend each have a [tex]\(\frac{1}{4}\)[/tex] probability of winning.
