Let's solve this problem step-by-step.
First, let's recall the concept of expected value. The expected value (EV) is a measure of the center of a probability distribution, giving us an idea of what we can expect on average from a random process.
For each game, we are given the probabilities for three outcomes: losing \[tex]$2, winning \$[/tex]1, and winning \[tex]$4. The expected value is computed as follows:
\[
EV = \sum ( \text{probability of outcome} \times \text{payout of outcome} )
\]
Let's compute the expected value for each game:
### Game 1:
The probabilities and payouts for Game 1 are:
- Probability of losing \$[/tex]2: [tex]\(0.55\)[/tex]
- Probability of winning \[tex]$1: \(0.20\)
- Probability of winning \$[/tex]4: [tex]\(0.25\)[/tex]
Calculating the expected value:
[tex]\[
EV_1 = (0.55 \times (-2)) + (0.20 \times 1) + (0.25 \times 4) = -1.1 + 0.2 + 1 = 0.1
\][/tex]
### Game 2:
The probabilities and payouts for Game 2 are:
- Probability of losing \[tex]$2: \(0.15\)
- Probability of winning \$[/tex]1: [tex]\(0.35\)[/tex]
- Probability of winning \[tex]$4: \(0.50\)
Calculating the expected value:
\[
EV_2 = (0.15 \times (-2)) + (0.35 \times 1) + (0.50 \times 4) = -0.3 + 0.35 + 2 = 2.05
\]
### Game 3:
The probabilities and payouts for Game 3 are:
- Probability of losing \$[/tex]2: [tex]\(0.20\)[/tex]
- Probability of winning \[tex]$1: \(0.60\)
- Probability of winning \$[/tex]4: [tex]\(0.20\)[/tex]
Calculating the expected value:
[tex]\[
EV_3 = (0.20 \times (-2)) + (0.60 \times 1) + (0.20 \times 4) = -0.4 + 0.6 + 0.8 = 1.0
\][/tex]
Summary of Expected Values:
- Expected value for Game 1: [tex]\(0.1\)[/tex]
- Expected value for Game 2: [tex]\(2.05\)[/tex]
- Expected value for Game 3: [tex]\(1.0\)[/tex]
### Decision:
b. If Tanya decides she will play the game, which game should she choose? Explain.
Since Tanya can only choose one game and she wants to maximize her expected winnings, she should choose the game with the highest expected value. Comparing the expected values calculated:
- [tex]\(EV_1 = 0.1\)[/tex]
- [tex]\(EV_2 = 2.05\)[/tex] <-- highest
- [tex]\(EV_3 = 1.0\)[/tex]
Thus, Tanya should choose Game 2. This game has the highest expected value of 2.05, which means on average, she can expect to gain \$2.05 every time she plays this game.
