Expected Value
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. You're level 1. You can read and practice here now.
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
- 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
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
- Zero means chance × gain equals (1 − chance) × loss.
- So the chance is the loss divided by the gain plus the loss.
worked 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
- 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
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
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.
practice
Expected value: two outcomes
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
- 0.8 × $1,100 − 0.2 × $1,000 = $880 − $200 = $680.
Expected value: three scenarios
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
- 0.55 × $500 + 0.3 × $0 − 0.15 × $600 = $275 + $0 − $90 = $185.
- The probabilities add to 100%, so this is the average result over many such projects.
Break-even probability
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%
- Set p × 200 = (1 − p) × 1,400, so p = 1,400 ÷ (200 + 1,400) = 1,400 ÷ 1,600 = 87.5%.
Average versus survival
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?
- B has the higher expected value, but its bad case could sink the business — A may be wiser
- They're the same because the average works out
- A has the higher expected value
- 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
- B's expected value is 0.6 × $3,000 − 0.4 × $1,700 = $1,120, above A's $800.
- But a $1,700 loss would leave cash at −$1,100. Expected value ignores whether you survive the bad outcome.