Making decisions under uncertainty
Many decisions involve chance: buying insurance, choosing a medical treatment, starting a business, packing an umbrella. Probability helps you weigh options.
Risk = probability × impact
Risk combines how likely something is (probability) with how bad it would be.
| Event | Probability | Impact | Risk level |
|---|---|---|---|
| Phone screen cracks | moderate | small–medium | moderate |
| House fire | low | very high | worth insuring against |
| Light rain on a hike | high | low | pack a jacket |
Expected value for decisions
Compare choices by their expected value.
Example: Should a farmer plant crop A or B?
| Good weather (P = 0.7) | Bad weather (P = 0.3) | Expected value | |
|---|---|---|---|
| Crop A | $50,000 | $10,000 | 0.7(50k) + 0.3(10k) = $38,000 |
| Crop B | $40,000 | $30,000 | 0.7(40k) + 0.3(30k) = $37,000 |
Crop A has the higher expected value, but Crop B is safer (its worst case is better). The choice depends on risk tolerance.
Decision trees
A decision tree maps choices (squares), chance events (circles), probabilities and payoffs. Work backwards from the ends, calculating expected values at each chance point, then choose the best branch.
Insurance decisions
Insurance usually has negative expected value for the buyer (the company must make a profit), yet it’s often wise because it protects against losses you couldn’t afford.
Health and safety risks
Use absolute risk to make sense of headlines.
Example: "Doubles the risk!" From 1 in 100,000 to 2 in 100,000 is a 100% relative increase but only 0.001 percentage points absolutely.
Comparing everyday risks
People often misjudge risk:
- Rare, dramatic events (plane crashes, shark attacks) feel more dangerous than common ones (car accidents, falls), even though the common ones cause far more harm.
- Familiar risks feel safer than new ones.
Looking at the actual probabilities helps correct these instincts.
Other factors
Good decisions also consider:
- Values and ethics
- Worst-case outcomes
- Reversibility
- Variability (standard deviation) of outcomes, not just the average
Worked examples
Example 1: A $40 warranty covers a $400 repair with a 6% chance. Expected repair cost without the warranty? 0.06 × 400 = $24. The warranty costs more than the expected repair.
Example 2: Rain forecast is 70%. Carrying an umbrella is a small cost; getting soaked is unpleasant. Decision? Take the umbrella.
Example 3: Investment A: E = $1,000, very variable. Investment B: E = $900, steady. Which suits a cautious investor? B.
Common mistakes
- Deciding on expected value alone, ignoring the worst case.
- Being swayed by relative risk headlines.
- Overestimating rare, dramatic risks.
Try it yourself
- A game: 20% chance to win $50, 80% to lose $10. Expected value?
- A $15 extended warranty covers a $200 repair with a 5% chance. Expected repair cost?
- Risk goes from 3 in 1,000 to 4 in 1,000. Relative and absolute increase?
- Why might someone choose a lower expected value option?
Answers: 1) $2 2) $10 3) about 33%; 0.1 percentage points 4) to reduce risk or avoid a bad worst case
Quick recap
- Risk combines probability and impact.
- Use expected value and decision trees to compare options.
- Consider risk tolerance, worst cases and absolute risk.