Answer :
To determine the unknown payoff for outcome 6 that makes this game fair (i.e., expected value equal to \[tex]$0), we need to rely on the concept of expected value. Since the game is fair, the expected value of the payoffs should sum to 0.
First, let's set up the problem with the known information:
| Outcome | Probability | Payoff |
|----------|--------------|--------|
| 1 | 1/6 | \$[/tex] 3 |
| 2 | 1/6 | \[tex]$ 5 | | 3 | 1/6 | -\$[/tex] 10 |
| 4 | 1/6 | \[tex]$ 11 | | 5 | 1/6 | \$[/tex] 3 |
| 6 | 1/6 | \[tex]$ x | Where \( x \) is the unknown payoff for outcome 6. The expected value (E) is calculated by summing the products of outcomes and their probabilities: \[ E = \sum(\text{Probability of Outcome} \times \text{Payoff}) \] For a fair game, the expected value is 0. Hence, \[ E = \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 5\right) + \left(\frac{1}{6} \times (-10)\right) + \left(\frac{1}{6} \times 11\right) + \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times x\right) = 0 \] We can simplify this: \[ \frac{1}{6} \left( 3 + 5 + (-10) + 11 + 3 + x \right) = 0 \] \[ \frac{1}{6} \left( 12 + x \right) = 0 \] \[ 12 + x = 0 \] \[ x = -12 \] However, based on the solution provided, it turns out the game's constructed expected value needs to be re-evaluated. Among the given answers: A. \$[/tex] -11
B. \[tex]$ 12 C. \$[/tex] 9
D. \$ 10
Based on additional considerations or contextual constraints we derived from solving it computationally, none of these alternatives yield exactly -12. Therefore when we sum these considerations, it concludes that:
The correct answer is actually not among the options provided.
Thus, the question indicates selecting from these values implies some miscalculated payoff outputs always summing incorrect expected points per mentioned constraints.
Given the context provided led back to None as correct output which means no answers match exact solution correctly amongst provided variables.
| 2 | 1/6 | \[tex]$ 5 | | 3 | 1/6 | -\$[/tex] 10 |
| 4 | 1/6 | \[tex]$ 11 | | 5 | 1/6 | \$[/tex] 3 |
| 6 | 1/6 | \[tex]$ x | Where \( x \) is the unknown payoff for outcome 6. The expected value (E) is calculated by summing the products of outcomes and their probabilities: \[ E = \sum(\text{Probability of Outcome} \times \text{Payoff}) \] For a fair game, the expected value is 0. Hence, \[ E = \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 5\right) + \left(\frac{1}{6} \times (-10)\right) + \left(\frac{1}{6} \times 11\right) + \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times x\right) = 0 \] We can simplify this: \[ \frac{1}{6} \left( 3 + 5 + (-10) + 11 + 3 + x \right) = 0 \] \[ \frac{1}{6} \left( 12 + x \right) = 0 \] \[ 12 + x = 0 \] \[ x = -12 \] However, based on the solution provided, it turns out the game's constructed expected value needs to be re-evaluated. Among the given answers: A. \$[/tex] -11
B. \[tex]$ 12 C. \$[/tex] 9
D. \$ 10
Based on additional considerations or contextual constraints we derived from solving it computationally, none of these alternatives yield exactly -12. Therefore when we sum these considerations, it concludes that:
The correct answer is actually not among the options provided.
Thus, the question indicates selecting from these values implies some miscalculated payoff outputs always summing incorrect expected points per mentioned constraints.
Given the context provided led back to None as correct output which means no answers match exact solution correctly amongst provided variables.
