Computing expected value in a game of chance

List every payoff, multiply by its probability, sum. The sign of the answer tells you whether the game is gain (positive), lose (negative), or break-even (zero).

01 Mini-Lesson Video

▸ VIDEO COMING SOON

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

02 What This Looks Like on ALEKS

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

Debra's spinner (6 slices) v1

Debra spins a spinner with 6 equal-sized slices numbered 1 through 6. She wins $1 if the spinner stops on 1, $3 on 2, $5 on 3, $7 on 4. She loses $8 if it stops on 5 or 6.

(a) Find the expected value per spin.
(b) In the long run, can Debra expect to gain, lose, or break even?

Roll-and-win die v2

You roll a fair six-sided die. You win $10 if you roll a 6, $2 if you roll a 4 or 5, and you lose $3 on a 1, 2, or 3.

(a) Find the expected value per roll.
(b) Gain, lose, or break even in the long run?

Card draw payoff v3

A card is drawn from a standard 52-card deck. You win $20 if you draw an ace, $5 if you draw a face card, −$1 (you lose a dollar) on any other card.

(a) Find the expected value per draw.
(b) Gain, lose, or break even?

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 How to Solve It

E(X) = Σ x · P(x)
multiply each payoff by its probability, sum across all outcomes

Expected value is the long-run average payout per trial. The sign tells you who has the edge.

List outcomes, payoffs, probabilities.

Six equal slices → each probability is 1/6.

+$1, +$3, +$5, +$7 \cdot\cdot\cdot each at 1/6 -$8, -$8 \cdot\cdot\cdot each at 1/6

Multiply each payoff by 1/6, then sum.

E(X) = (1 + 3 + 5 + 7 - 8 - 8) / 6 = 0 / 6 = $0

Interpret the sign.

E(X) = $0 means the game is fair — Debra breaks even in the long run. Some spins she wins, others she loses, but averaged across many spins her balance stays where she started.

▸ COMMON SLIPS(1) Forgot one of the losing outcomes. Two slices lose $8 (slices 5 AND 6) — that's −$16/6 total contribution from losses. Missing one inflates E by $8/6 ≈ +$1.33. (2) Used the wrong probability for unequal outcomes. If the spinner had unequal slices, each outcome would have its own P(x). Here all six slices are equal → 1/6 each. Check that. (3) Confused E = 0 with "impossible to win." Break-even doesn't mean you never win; it means wins and losses balance out on average.

04 Practice — Step by Step

Try the roll-and-win die. Same recipe: list outcomes, multiply, sum.

Roll-and-win expected value.

Fair die: win $10 on a 6, $2 on a 4 or 5, lose $3 on a 1, 2, or 3. What is E(X)? (Decimal to two places.)

E(X) = $

▸ NICE WORK

You've walked through the whole problem.

That's the move. ALEKS will give you a different version with different numbers — but the steps are the same.

Q4 Q6