How do you compute the expected value of a decision with outcomes {v1, v2} and probabilities {p1, p2}?

How do you compute the expected value of a decision with outcomes {v1, v2} and probabilities {p1, p2}?

• A. EV = p1*v1
• B. EV = p2*v2
• C. EV = p1*v2 + p2*v1
• D. EV = p1*v1 + p2*v2

Answer: (D)

Calculating the expected value means averaging the possible results while weighting each by its probability. With two outcomes, that weighted average is EV = p1*v1 + p2*v2. This reflects that you gain v1 with probability p1 and v2 with probability p2, and over many trials the average outcome converges to this weighted sum. The other forms miss an outcome or pair the wrong values with the wrong probabilities: p1*v1 only accounts for the first outcome, p2*v2 only for the second, and p1*v2 + p2*v1 swaps the outcomes with the probabilities, which doesn’t match what actually happens.

Calculating the expected value means averaging the possible results while weighting each by its probability. With two outcomes, that weighted average is EV = p1v1 + p2v2. This reflects that you gain v1 with probability p1 and v2 with probability p2, and over many trials the average outcome converges to this weighted sum. The other forms miss an outcome or pair the wrong values with the wrong probabilities: p1v1 only accounts for the first outcome, p2v2 only for the second, and p1v2 + p2v1 swaps the outcomes with the probabilities, which doesn’t match what actually happens.