courses › Thinking

Expected value

level 21 course

Weigh each outcome by its chance before you decide.

In your head, jot if needed · no calculator why?

Opens at level 21.

Builds on: Chance basics (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Expected value is the average result per try if you could repeat a choice many times: each outcome times its chance, added up. A game that costs $5 and pays $40 one time in ten returns $4 on average, so it loses $1 a play.

It doesn't predict one try; it tells you which choice wins over many. For a loss you couldn't afford, protection can still be worth a small average cost.

Techniques

Weigh the prize by its chance

You pay to play and win a prize with some chance.

Average winnings: the prize × the chance of winning.
Take off the cost to play, which you pay every time.
A negative result is your average loss per play.

worked example

Example: It costs $10 to play. You win $45 with a 20% chance, otherwise nothing. What is your expected profit per play? (Use a minus sign for a loss.)

Average winnings: $45 × 20% = $9.
Minus the $10 cost: $9 − $10 = −$1.

Answer: −$1

Stake over total payout

The chance of winning that makes a bet break even.

Break even means chance × total payout equals the stake.
So the chance is the stake divided by the total payout.
Use the total you get back, stake included, not the profit.

worked example

Example: A $10 entry fee returns $40 in total (your fee included) if you win, and nothing otherwise. What chance of winning makes it break even? Give a percent.

Chance × $40 = $10.
$10 ÷ $40 = 0.25 = 25%.

Answer: 25%

Price against expected payout

Deciding on insurance, a warranty or a protection plan.

Expected payout: the chance of the loss × what the plan pays.
Compare it with the price. The gap is your average gain or loss.
Price above the expected payout: skip it, unless the loss would really hurt.

worked example

Example: A $50 warranty covers a $300 repair that has a 10% chance of being needed. On average, how much more does the warranty cost than it pays out?

Expected payout: 10% × $300 = $30.
Price minus payout: $50 − $30 = $20.

Answer: $20

Tips by skill

TipIs the bet worth it?: Average winnings are the prize times the chance. Subtract the cost to play; the result can be negative.
TipBreak-even chance: Divide the stake by the total payout, stake included. At that chance the average win equals the stake.
TipInsurance & warranties: Multiply the chance by the covered loss, then compare with the price. The gap is the average gain or loss.

Watch out for

Using the prize minus the cost, as if you always won. Weight the prize by how rarely you win.
Dividing the stake by the profit instead of the total payout. A $20 bet that returns $120 breaks even at 20 ÷ 120, not 20 ÷ 100.
Comparing a warranty's price with the full repair cost instead of the repair's expected cost.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Is the bet worth it? par 20 s not tried yet
worked example
It costs $18 to play. You win $200 with a 1-in-20 chance, otherwise nothing. What is your expected profit per play? (Use a minus sign for a loss.)

Answer: -$8.00
1. Expected winnings: $200 × 1/20 = $10.
2. Minus the $18 cost: −$8 per play on average.
Break-even chance par 20 s not tried yet
worked example
A $10 bet pays back $20 in total (including your $10) if it wins, and nothing if it loses. What win probability makes it break even? Give a percent.

Answer: 50%
1. You break even when chance × payout = stake: p × $20 = $10.
2. So p = 10 ÷ 20 = 50%.
Insurance & warranties par 25 s not tried yet
worked example
Phone insurance costs $280 a year and covers a $900 screen replacement. There is a 25% chance you'll need one this year. What does expected value say?
1. Buy — on average it pays out $55 more than it costs
2. Buy — it saves $620 on average
3. Skip — you lose the whole $280 on average
4. Skip — on average it costs $55 more than it pays out
Answer: Skip — on average it costs $55 more than it pays out
1. Expected payout: 25% × $900 = $225.
2. It costs $280, so on average you lose $55. (It can still make sense if the loss would really hurt.)

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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