To determine if the game is fair, we need to calculate the probabilities of each player winning. We start by listing all the possible outcomes when tossing two coins.
The possible outcomes are:
1. Heads, Heads (HH)
2. Heads, Tails (HT)
3. Tails, Tails (TT)
4. Tails, Heads (TH)
Next, we determine the winning conditions for both you and your friend:
- You win if both coins land on heads (HH).
- Your friend wins if both coins land on tails (TT).
- Nobody wins if the coins land on heads and tails (HT or TH).
Now, we calculate the probabilities of these events:
1. Probability of Winning for You:
- There is only one outcome where you win: (HH).
- The probability of this outcome occurring is the number of favorable outcomes divided by the total number of possible outcomes.
[tex]\[
\text{Probability you win} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{4} = 0.25
\][/tex]
2. Probability of Winning for Your Friend:
- There is only one outcome where your friend wins: (TT).
- The probability of this outcome occurring is the number of favorable outcomes divided by the total number of possible outcomes.
[tex]\[
\text{Probability your friend wins} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{4} = 0.25
\][/tex]
3. Probability that Nobody Wins:
- There are two outcomes where nobody wins: (HT) and (TH).
- The probability of these outcomes is the sum of their individual probabilities.
[tex]\[
\text{Probability nobody wins} = \frac{2}{4} = 0.5
\][/tex]
Since both you and your friend each have a probability of [tex]\( \frac{1}{4} \)[/tex] (or 0.25) of winning, 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.
