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Sagot :
Let's break this problem into parts and solve it step-by-step.
a. What is the expected value for playing each game?
To calculate the expected value for each game, we use the formula for expected value:
[tex]\[ \text{Expected Value} = \sum (\text{probability} \times \text{payoff}) \][/tex]
We'll calculate this for each game, using the probabilities and the payoffs provided.
Game 1:
- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.55\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.20\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.25\)[/tex]
[tex]\[ \text{Expected Value of Game 1} = (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 4) \][/tex]
Game 2:
- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.15\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.35\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.50\)[/tex]
[tex]\[ \text{Expected Value of Game 2} = (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 4) \][/tex]
Game 3:
- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.20\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.60\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.20\)[/tex]
[tex]\[ \text{Expected Value of Game 3} = (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 4) \][/tex]
Now, evaluating these expressions:
For Game 1:
[tex]\[ (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 4) = -1.1 + 0.2 + 1.0 = 0.1 \][/tex]
So, the expected value for Game 1 is [tex]\(0.1\)[/tex].
For Game 2:
[tex]\[ (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 4) = -0.3 + 0.35 + 2.0 = 2.05 \][/tex]
So, the expected value for Game 2 is [tex]\(2.05\)[/tex].
For Game 3:
[tex]\[ (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 4) = -0.4 + 0.6 + 0.8 = 1.0 \][/tex]
So, the expected value for Game 3 is [tex]\(1.0\)[/tex].
In summary, the expected values are:
- Game 1: [tex]\(0.1\)[/tex]
- Game 2: [tex]\(2.05\)[/tex]
- Game 3: [tex]\(1.0\)[/tex]
b. If Tanya decides she will play the game, which game should she choose? Explain.
To determine which game Tanya should choose, we look at the expected values calculated:
- Expected value of Game 1: [tex]\(0.1\)[/tex]
- Expected value of Game 2: [tex]\(2.05\)[/tex]
- Expected value of Game 3: [tex]\(1.0\)[/tex]
Since Tanya wants to maximize her expected winnings, she should choose the game with the highest expected value. Among the three games, Game 2 has the highest expected value of [tex]\(2.05\)[/tex].
Therefore, Tanya should choose Game 2.
a. What is the expected value for playing each game?
To calculate the expected value for each game, we use the formula for expected value:
[tex]\[ \text{Expected Value} = \sum (\text{probability} \times \text{payoff}) \][/tex]
We'll calculate this for each game, using the probabilities and the payoffs provided.
Game 1:
- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.55\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.20\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.25\)[/tex]
[tex]\[ \text{Expected Value of Game 1} = (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 4) \][/tex]
Game 2:
- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.15\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.35\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.50\)[/tex]
[tex]\[ \text{Expected Value of Game 2} = (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 4) \][/tex]
Game 3:
- Probability of losing [tex]\(-\$2\)[/tex]: [tex]\(0.20\)[/tex]
- Probability of winning [tex]\(\$1\)[/tex]: [tex]\(0.60\)[/tex]
- Probability of winning [tex]\(\$4\)[/tex]: [tex]\(0.20\)[/tex]
[tex]\[ \text{Expected Value of Game 3} = (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 4) \][/tex]
Now, evaluating these expressions:
For Game 1:
[tex]\[ (0.55 \times -2) + (0.20 \times 1) + (0.25 \times 4) = -1.1 + 0.2 + 1.0 = 0.1 \][/tex]
So, the expected value for Game 1 is [tex]\(0.1\)[/tex].
For Game 2:
[tex]\[ (0.15 \times -2) + (0.35 \times 1) + (0.50 \times 4) = -0.3 + 0.35 + 2.0 = 2.05 \][/tex]
So, the expected value for Game 2 is [tex]\(2.05\)[/tex].
For Game 3:
[tex]\[ (0.20 \times -2) + (0.60 \times 1) + (0.20 \times 4) = -0.4 + 0.6 + 0.8 = 1.0 \][/tex]
So, the expected value for Game 3 is [tex]\(1.0\)[/tex].
In summary, the expected values are:
- Game 1: [tex]\(0.1\)[/tex]
- Game 2: [tex]\(2.05\)[/tex]
- Game 3: [tex]\(1.0\)[/tex]
b. If Tanya decides she will play the game, which game should she choose? Explain.
To determine which game Tanya should choose, we look at the expected values calculated:
- Expected value of Game 1: [tex]\(0.1\)[/tex]
- Expected value of Game 2: [tex]\(2.05\)[/tex]
- Expected value of Game 3: [tex]\(1.0\)[/tex]
Since Tanya wants to maximize her expected winnings, she should choose the game with the highest expected value. Among the three games, Game 2 has the highest expected value of [tex]\(2.05\)[/tex].
Therefore, Tanya should choose Game 2.
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