Probability and Expected Value
The basic rules of probability an investor needs, how to weigh uncertain outcomes into an expected value, why a good bet can still ruin you if it's too large, and how evidence should change your estimate.
The short version
Investing means choosing without knowing the outcome, and probability is the tool for doing that carefully. An expected value weighs each possible result by its chance, which gives a single figure for comparing uncertain choices. It has limits. An expected value describes the average of many tries, and you may only get one. A bet with a positive expected value will still wipe you out if you stake too much on each try. And a signal that's right 90% of the time can be wrong most of the times it fires, if the thing it detects is rare. Thinking in probabilities won't tell you what will happen. It makes you say what you believe and how sure you are.
By the end of this lesson, you'll be able to
- Apply the rules for combining probabilities, including the chance of at least one event over many periods
- Calculate an expected value and explain what it does and doesn't tell you
- Show how the size of a bet and the rarity of an event change what a favourable probability is worth
Why this matters
Nobody knows what a share will be worth in five years. Every investing decision is a choice among uncertain outcomes, and the only question is whether the uncertainty is handled carefully or ignored. Scenario Analysis: Bull, Base, and Bear Cases weighted three valuations by their chances. This lesson covers the reasoning underneath that, starting from the basics.
Three rules
A probability is a number from 0 to 1, or 0% to 100%, that says how likely something is. Three rules cover most of what an investor needs.
- The chances of all possible outcomes add up to 100%. If you've listed three scenarios and given them 50%, 40%, and 30%, something is wrong.
- For independent events, the chance that all happen is the product of their chances. Independent means one doesn't affect the other. Two coin flips both landing heads: 0.5 × 0.5 = 25%.
- The chance of at least one is 100% minus the chance of none. This is the one that's easiest to underestimate.
Suppose there's a 5% chance each year of a serious setback, and each year is independent of the others.
| Years | Chance of avoiding it every year | Chance of at least one |
|---|---|---|
| 1 | 95% | 5% |
| 10 | 0.9510 = 60% | 40% |
| 20 | 0.9520 = 36% | 64% |
| 30 | 0.9530 = 21% | 79% |
Something that's unlikely in any single year becomes likely over an investing lifetime. A plan that works only if a 1-in-20 event never happens isn't a safe plan over 30 years.
The second rule needs care. It applies only when events really are independent. Ten shares don't each have a separate chance of falling in a crash, because a crash is one event that reaches all of them. Correlation, Plainly covers that.
Expected value
An Expected value. The average outcome of an uncertain choice, with each possible result weighted by its probability. It describes what you'd average over many tries, not what any one try will give. Glossary is the average outcome, with each possibility weighted by its probability. Multiply each outcome by its chance and add the results.
A company is waiting for a ruling on its main product. Its shares trade at $30.
| Outcome | Probability | Share value | Probability × value |
|---|---|---|---|
| Approved | 30% | $90 | $27.00 |
| Rejected | 70% | $15 | $10.50 |
| Expected value | 100% | $37.50 |
The expected value is $37.50, which is 25% above the price. On these numbers the bet is favourable. It's also a bet that loses half its value seven times out of ten. Both statements are true, and the expected value alone gives you only the first.
Two things about the figure $37.50. The share will never be worth that: it'll be worth $90 or $15. And the whole calculation rests on the 30%, which is somebody's estimate. An expected value is only as good as the probabilities that go into it.
The rest of this lesson is for members
Still to come in this lesson: What expected value leaves out, Rare things and imperfect signals, Using it without fooling yourself, the key takeaways, and 2 quiz questions with explanations.
Membership opens every lesson in the curriculum, the capstone projects, the interactive tools, and the members-only channels in the Hyperion Entities Discord. The first lesson of every module stays free.
Go deeper
- Distributions, Standard Deviation, and Fat TailsMath and Statistics for Investors (depth track)
- Scenario Analysis: Bull, Base, and Bear CasesValuation and Modeling (depth track)
- Valuation Under UncertaintyValuation and Modeling (depth track)
- Arithmetic and Geometric AveragesMath and Statistics for Investors (depth track)