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 Worked Example Video

Every step of Computing expected value in a game of chance explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

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)

Debra spins a 6-slice spinner. Payoffs by slice: $1, $3, $5, $7 on slices 1-4; −$8 on slice 5 and slice 6.

Find E(X) per spin. Type just the number (no $ sign).

E(X) = $

Roll-and-win die

Roll a fair die. Win $10 on a 6, $2 on 4 or 5, lose $3 on 1, 2, or 3.

Find E(X) per roll. Round to two decimal places.

E(X) = $

Card draw payoff

Draw a card from a 52-card deck. Win $20 on an ace (4 cards), $5 on a face card (12 cards), lose $1 on any other card (36 cards).

Find E(X) per draw.

E(X) = $

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

Sign tells the story + = gain, − = lose, 0 = break even

Expected value is the long-run average payout per trial. Multiply each payoff by its probability, sum across all outcomes. Losses get negative signs; gains stay positive. The sign of the sum tells you who has the edge. See it live: this exact three-payoff die game in the Expected Value Lab — E(X) = +$1.00/roll →

Common slips

(1) Forgot to negate losses. “Loses $8” enters the formula as -8, not 8. That flip changes the sign of E(X) entirely. (2) Wrong probability weights. On a 6-sided die, “5 or 6” is 2/6 (not 1/6). Each outcome’s weight must reflect how many of its favorable slices exist. (3) Interpreted the answer as a per-outcome, not per-trial. E(X) is the long-run average per trial, not per winning slot.

04 Practice — Step by Step

Practice this problem step by step

Roll a fair 6-sided die. Win $9 on a 6, win $3 on 4 or 5, lose $3 on 1, 2, or 3. Find the expected value per roll and interpret it.

Step 1

Weight the top payoff

P(roll 6) = 1/6. Payoff = $9. Compute 9 × (1/6).

= $

Step 2

Sum all three weighted payoffs

Add: 9(1/6) + 3(2/6) + (-3)(3/6) = 1.5 + 1 − 1.5. Type just the number.

E(X) = $

Step 3

Interpret the sign

E(X) = +$1 per roll. In the long run, do you expect to:

Q4 Q6

Computing expected value in a game of chance

You walked expected value end to end.