Expected Value

Learn expected value, expected net gain.

Lesson 2 of 32 parts Practice Flashcards

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Expected Value

Objective: Explain expected value, apply the stated method, and check its conditions.

A discrete expected value is E[X]=ΣxP(X=x). It is a long-run weighted average, not a guarantee for a single trial and not necessarily an attainable outcome. Units match the variable; nonlinear transformations generally cannot be applied directly to an expectation. The expected value need not occur in any single trial.

Formulas

$E[X]=ΣxP(X=x)$

Worked examples

Try itA synthetic game pays $42 with probability 1/7 and $0 otherwise. Find expected payout.Show answer

Step 1: Expected value is a probability-weighted average. Step 2: Compute (42)(1/7)+0(1−1/7). Step 3: Expected payout is $6; it need not be a possible single-play payout.

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Expected Value

Objective: Explain expected value, apply the stated method, and check its conditions.

Formulas

$E[X]=ΣxP(X=x)$

Worked examples

Try itA synthetic game pays $42 with probability 1/7 and $0 otherwise. Find expected payout.Show answer

Step 1: Expected value is a probability-weighted average. Step 2: Compute (42)(1/7)+0(1−1/7). Step 3: Expected payout is $6; it need not be a possible single-play payout.

Expected Net Gain

Objective: Explain expected net gain, apply the stated method, and check its conditions.

Expected net gain equals expected gross payout minus a fixed entry cost. Linearity of expectation permits subtracting that cost without changing outcome probabilities. A positive gross expected payout can coexist with a negative expected net gain. Include the cost before judging expected gain.

Formulas

E[net]=E[payout]−fixed cost

Worked examples

Try itA synthetic game pays $42 with probability 1/7 and $0 otherwise, and costs $7 to play. Find expected net gain.Show answer

Step 1: Expected gross payout is (42)(1/7)=6. Step 2: Subtract the entry cost 7. Step 3: Expected net gain is $-1.

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