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All courses Math III · S-MD.7 55 of 55
Analyze decisions and strategies using probability concepts in more complex settings.

Find the expected value by multiplying each outcome by its probability and adding

Problem
What is the expected value of this decision: win $10 with probability 0.2 and lose $2 with probability 0.8?
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Hint

Multiply each outcome by its probability, using a negative value for the loss.

Expected value is a weighted average: sum of \(probability × outcome\) across all outcomes.

Solution walkthrough

01

Write the expected-value structure

\[EV=\text{sum}~\text{of}~\text{probability}*\text{payoff}\]

Each possible monetary outcome is weighted by its probability.

02

Represent both signed outcomes

\[\text{win}~\text{payoff}=10~\text{with}~\text{probability}~0.2;~\text{loss}~\text{payoff}=-2~\text{with}~\text{probability}~0.8\]

A loss must enter the expectation as a negative payoff.

03

Compute the weighted sum

\[EV=0.2(10)+0.8(-2)=2-1.6=0.40\]

The expected gain contribution is two dollars and the expected loss contribution is negative one dollar sixty cents.

04

State the expected value

\[0.2(10)+0.8(-2)=0.40~\text{dollars}\]

Over many comparable decisions, the average net payoff is forty cents per play.

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Another way

  1. Out of ten representative plays, expect two wins worth 20 dollars and eight losses worth -16 dollars, for 4 dollars total or 0.40 dollars per play.

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Common mistake

Do not use positive 2 dollars for the loss outcome; its payoff is -2 dollars.