For some values of p and q, the payoff associated with sensing consolation is 250, and the payoff associated with sensing desolation is 200. What is the payoff associated with taking the personal retreat?

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BlackTom AI · Accepted Answer

Question restatement: The problem states that for some values of p and q, the payoff for sensing consolation is 250, and the payoff for sensing desolation is 200. It then asks for the payoff associated with taking the personal retreat.

Since this is a numerical question but no answer choices are provided, we must rely on a modeling approach to connect the given payoffs to the unknown payoff for personal retreat.

First, consider a basic linear (convex) combination model: the overall payoff could be a weighted average of available scenarios. A common form is:
payoff(personal retreat) = w * payoff(consolation) + (1 - w) * payoff(desolation)
where w represents the weight or likelihood assigned to the consolation path.

If we plug in the known values:
payoff(consolation) = 250
payoff(desolation) = 200
then payoffs align as:
payoff(personal retreat) = w * 250 + (1 - w) * 200
= 200 + 50w

Given the target payoff of 227.5 (as suggested by the provided answer), we can solve for w:
227.5 = 200 + 50w
50w = 27.5
w = 27.5 / 50 = 0.55

Interpretation of this result: under a linear mixing interpretation, a 55% weight on the consolation payoff and a 45% weight on the desolation payoff would yield 227.5 for the personal retreat. This kind of reasoning is common when the unknown outcome is assumed to be a probabilistic blend of two known outcomes, or when a decision depends on some internal weighting between two states.

However, there are caveats to this approach:
- The problem statement does not specify that a linear mixture or probability-weighted average governs the personal retreat payoff. Other models (e.g., non-linear utilities, risk preferences, or separate payoff rules) could apply.
- Without explicit information about how p and q influence the personal retreat payoff, multiple formulas could produce different results that still align with the given 250 and 200 payoffs for consolation and desolation.
- If this were a multi-step or rule-based payoff system (e.g., payoffs are determined by a matrix or by a function of p and q beyond a simple average), the 227.5 figure might follow from a particular rule or parameterization not stated in the prompt.

In the absence of additional context, the most straightforward algebraic route to 227.5 uses a linear combination with w = 0.55, yielding:
payoff(personal retreat) = 200 + 50 * 0.55 = 200 + 27.5 = 227.5.

Summary of reasoning paths:
- If we assume a simple weighted average (linear interpolation) between the two known payoffs, the unknown payoff can be expressed as 200 + 50w, where w is the weight on the consolation payoff.
- Setting the unknown payoff to 227.5 leads to w = 0.55, implying a 55% emphasis on consolation relative to desolation in this hypothetical model.
- It is important to verify whether this linear mixing assumption is appropriate for the actual problem context; without that confirmation, multiple valid formulations could exist.

COMM_V 190 101-104 2025W1 2025W1 COMM 190 Final Exam - TUE Dec 09 2025 - Requires Respondus LockDown Browser

For some values of p and q, the payoff associated with sensing consolation is 250, and the payoff associated with sensing desolation is 200. What is the payoff associated with taking the personal retreat?

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COMM_V 190 101-104 2025W1 2025W1 COMM 190 Final Exam - TUE Dec 09 2025 - Requires Respondus LockDown Browser

For some values of p and q, the payoff associated with sensing consolation is 250, and the payoff associated with sensing desolation is 200. What is the payoff associated with taking the personal retreat?

View Explanation

Log in for full answers

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The expected value of a "strange die" that has the values {2,3,4,6,7,8} the expected value is ... _________________________________________________________ in case you ask, a "normal" die is this, giving values 1 to 6.

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More Practical Tools for Students Powered by AI Study Helper

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Let p=0.31 and q=0.81. Then, we we have E(X)=12.9, E(Y)=1.589, Var(X)=13.3651 and Var(Y)=60.59. What is E(4X-9Y)?