Expected value
Weigh each outcome by its chance before you decide.
In your head, jot if needed · no calculator why?
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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.
more of the idea
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
- 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.
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
- 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.
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
- 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.
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
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.
practice
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Is the bet worth it?
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.)
- Expected winnings: $200 × 1/20 = $10.
- Minus the $18 cost: −$8 per play on average.
Answer: -$8.00
Break-even chance
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.
- You break even when chance × payout = stake: p × $20 = $10.
- So p = 10 ÷ 20 = 50%.
Answer: 50%
Insurance & warranties
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?
- Buy — on average it pays out $55 more than it costs
- Buy — it saves $620 on average
- Skip — you lose the whole $280 on average
- Skip — on average it costs $55 more than it pays out
- Expected payout: 25% × $900 = $225.
- It costs $280, so on average you lose $55. (It can still make sense if the loss would really hurt.)
Answer: Skip — on average it costs $55 more than it pays out