Computing expected value in a game of chance

Every outcome gets weighted by how often it happens. The number that comes out is what one play is worth on average, and its sign answers the second part.

01 Worked Example Video

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A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 Recognize the Question

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

One winning face

A fair six-sided die. You win $5 on a 1 and lose $0.50 on any of 2, 3, 4, 5 or 6.

What is the expected value? Round to the nearest cent.

A fair game

A fair six-sided die. You win $2 on a 1 and $2 on a 2, and lose $1 on any of 3, 4, 5 or 6.

What is the expected value? Round to the nearest cent.

Read the sign

A fair six-sided die. You win $6 on a 1 and lose $1 on any of 2, 3, 4, 5 or 6.

What is the expected value? Round to the nearest cent.

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

E = Σ (value × probability) every outcome, including the losing ones

losses enter as negative values this is where most sign errors live

E > 0 gain · E < 0 lose · E = 0 break even part (b) is just reading the sign

On a fair six-sided die every outcome has probability 1/6, so the expectation is the total of the outcome values divided by six. Grouping the losing faces is a shortcut: four faces at −$0.40 contribute 4 × (−0.40) ÷ 6 in one step.

Common slips

(1) Entering a loss as positive. Losing $0.40 is −0.40 in the sum. (2) Forgetting the faces that pay nothing. They still carry probability and still belong in the denominator. (3) Answering the total instead of the average. Expected value is per play, not across all six faces.

04 Practice — Step by Step

Weight every face, then read the sign

Nadia is playing a game of chance in which she rolls a number cube with sides numbered from 1 to 6. The cube is fair, so a side is rolled at random.

The game is this: Nadia rolls the number cube once. She wins $3 if a 1 is rolled and $1 if a 2 is rolled. She loses $0.40 if a 3, 4, 5 or 6 is rolled.

(a) Find the expected value of playing the game.
(b) What can Nadia expect in the long run, after playing the game many times?

Step 1

How many outcomes carry a loss?

Faces 3, 4, 5 and 6 each lose $0.40.

Step 2

The winnings side

$3 on a 1 and $1 on a 2, each with probability 1/6. What is (3 + 1) ÷ 6? Give four decimal places.

Step 3

The losses side

Four faces at $0.40 each: (4 × 0.40) ÷ 6. Give four decimal places.

Step 4

(a) The expected value

0.6667 − 0.2667, to the nearest cent.

Step 5

(b) What does that mean?

The expected value is +$0.40.

Q5 Q7

Computing expected value in a game of chance

The sign is the answer to part (b)