Weight Every Outcome by Its Probability

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A supplied fictional payoff model gives +8 points with probability 1/4 and −2 points with probability 3/4. Find expected payoff per trial. Candidate answers: (A) -0.5 points (B) 1 points (C) 1.5 points (D) 1/2 points.
• Independent check inputs: A supplied fictional payoff model gives +12 points with probability 1/4 and −3 points with probability 3/4. Find expected payoff per trial. Candidate answers: (A) -0.25 points (B) 1.5 points (C) 1.75 points (D) 3/4 points.

Learn the idea

Counting methods distinguish ordered roles from unordered groups. A probability model assigns chances to possible outcomes, and expected value or binomial counting combines those probabilities while preserving what a single trial can actually produce. Expected value is a probability-weighted average of modeled outcomes. Positive and negative payoffs both enter with their signs, and their probabilities must describe a complete valid distribution.

Here is the method on the supplied teaching case. Weight each outcome by its probability: 8/4−3(2)/4=2/4=1/2. Expected value is a long-run mean of this model, not a guaranteed single-trial payoff.

An expected value need not be a possible single outcome. This fictional points model is educational, not wagering advice; it describes a model’s average rather than guaranteeing a gain on any trial.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A supplied fictional payoff model gives +36 points with probability 1/4 and −9 points with probability 3/4. Find expected payoff per trial. Candidate answers: (A) 1.25 points (B) 3.25 points (C) 4.5 points (D) 9/4 points. Show the mathematical reason, not only the selected answer. Expected result: 9/4 points. Weight each outcome by its probability: 36/4−3(9)/4=9/4=9/4. Expected value is a long-run mean of this model, not a guaranteed single-trial payoff.

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