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Expected Value

level 27 course

Probability-weighted outcomes, and why the average isn't the whole story.

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

Opens at level 27.

Builds on: Probability 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 probability-weighted average of the outcomes: each outcome times its chance, added up, with losses counted as negative. It is what a decision is worth on average over many like it.

Check two things first. The chances must add to 100%. And know whether the outcomes are net, with the decision's cost already taken out; if they are, don't subtract the cost again.

Techniques

Weigh each outcome

Two or more outcomes, each with a chance.

List each outcome with its chance, and check the chances add to 100%.
Multiply each outcome by its chance, as a decimal.
Add the gains and subtract the losses.

worked example

Example: A bike shop's event has net outcomes: gain $2,000 (30% chance), break even (50% chance), or lose $1,000 (20% chance). What is the expected net value? (Use a minus sign if negative.)

0.3 × $2,000 = $600.
The loss: 0.2 × $1,000 = $200.
$600 + $0 − $200 = $400.

Answer: $400

Loss over the total swing

The chance of success that makes the expected value exactly zero.

Zero means chance × gain equals (1 − chance) × loss.
So the chance is the loss divided by the gain plus the loss.

worked example

Example: A tutoring service's project either earns $600 net or loses $200 net. What chance of success makes its expected value exactly zero?

Total swing: $600 + $200 = $800.
$200 ÷ $800 = 0.25 = 25%.
Check: 0.25 × $600 = $150, and 0.75 × $200 = $150.

Answer: 25%

Then check the bad case

Choosing between a safe option and a risky one with a higher average.

Work out the risky option's expected value.
Then take its loss from the cash on hand.
Below zero: the higher average may not be worth the risk.

worked example

Example: A café has $3,000 of cash. Option B has a 50% chance to gain $8,000 and a 50% chance to lose $5,000. How much cash is left after the bad case? (Use a minus sign if negative.)

B's expected value: 0.5 × $8,000 − 0.5 × $5,000 = $1,500.
Bad case: $3,000 − $5,000 = −$2,000.

Answer: −$2,000

Tips by skill

TipExpected value: two outcomes: Chance × gain minus chance × loss, with each chance as a decimal. The outcomes are already net of costs.
TipExpected value: three scenarios: Multiply each outcome by its chance and add, counting losses as negative. The chances should add to 100%.
TipBreak-even probability: Divide the loss by the total swing, gain plus loss. At that chance the expected value is zero.
TipAverage versus survival: Compare the expected values, then take B's loss from the cash on hand. Whether cash stays above zero decides it.

Watch out for

Averaging the outcomes as if each were equally likely. A $2,000 gain at 25% and a $200 loss at 75% is worth $350, not $900.
Subtracting the cost again when the outcomes are already net of it.
Dividing the loss by the gain for the break-even chance. For a $300 gain or $100 loss it is 100 ÷ 400, 25%, not 100 ÷ 300.

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.

Expected value: two outcomes par 18 s not tried yet
worked example
A project's net outcomes are a $1,100 gain with 80% probability or a $1,000 loss with 20% probability. What is the expected net value? (Use a minus sign if negative.)

Answer: $680.00
1. 0.8 × $1,100 − 0.2 × $1,000 = $880 − $200 = $680.
Expected value: three scenarios par 23 s not tried yet
worked example
Net outcomes: gain $500 (55% chance), break even (30% chance), or lose $600 (15% chance). What is the expected net value? (Use a minus sign if negative.)

Answer: $185.00
1. 0.55 × $500 + 0.3 × $0 − 0.15 × $600 = $275 + $0 − $90 = $185.
2. The probabilities add to 100%, so this is the average result over many such projects.
Break-even probability par 20 s not tried yet
worked example
A project either earns $200 net or loses $1,400 net. What chance of success makes its expected value exactly zero?

Answer: 87.5%
1. Set p × 200 = (1 − p) × 1,400, so p = 1,400 ÷ (200 + 1,400) = 1,400 ÷ 1,600 = 87.5%.
Average versus survival par 20 s not tried yet
worked example
The business has $600 of cash. Option A guarantees $800. Option B has a 60% chance to gain $3,000 and a 40% chance to lose $1,700. Which statement is right?
1. B has the higher expected value, but its bad case could sink the business — A may be wiser
2. They're the same because the average works out
3. A has the higher expected value
4. B is always better because its expected value is higher
Answer: B has the higher expected value, but its bad case could sink the business — A may be wiser
1. B's expected value is 0.6 × $3,000 − 0.4 × $1,700 = $1,120, above A's $800.
2. But a $1,700 loss would leave cash at −$1,100. Expected value ignores whether you survive the bad outcome.

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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