Stock market foundationsModule 6, lesson 26 of 29
Probability and expected value
How to think about probability, calculate an expected value, judge a decision separately from its outcome, and why a small chance of a large loss matters so much.
In this lesson
- Express a probability as a fraction, a decimal and a percentage, and work out the chance of something happening at least once.
- Calculate the expected value of a bet or decision with several outcomes.
- Explain why a positive expected value does not guarantee a good result, especially when losses can be large.
- Separate the quality of a decision from the luck of its outcome.
Builds on
- Percentages, percentage change and percentage pointsProbabilities and returns are both written as percentages, and repeated changes multiply.
- The risks of investing in sharesThis lesson puts numbers on the risks described there.
On this page
In simple terms
A probability describes how likely something is. It runs from 0 (it will not happen) to 1 (it certainly will), and is often written as a percentage: a 25% chance is a probability of 0.25, or 1 in 4.
One useful way to think about it: if a situation could be repeated many times, the probability is roughly the share of times the event would happen. A fair coin lands heads about half the time over many tosses, even though any single toss is unpredictable.
Expected value combines probabilities with what is at stake. It is the average result you would get if you could repeat a decision many times.
Calculating expected value
Expected value = (outcome 1 × probability 1) + (outcome 2 × probability 2) + …
The probabilities of all the possible outcomes must add up to 1.
Worked example: A lottery-style bet
A ticket costs ₹100 and has a 1 in 1,000 chance of winning ₹50,000.
| Average winnings per ticket: 0.001 × 50,000 | ₹50 |
|---|---|
| Cost per ticket | −₹100 |
| Expected value per ticket | −₹50 |
Someone will win, and their outcome will be wonderful. But anyone who buys many tickets should expect to lose about half their money. Most lotteries, and many "high return" schemes, are built this way.
The chance of something happening at least once
When something is repeated, small chances add up quickly. The easiest way to work it out is through the chance that it never happens.
Chance of at least one occurrence in n independent tries = 1 − (chance it does not happen in one try)n
A strategy that works 90% of the time sounds safe. Used ten times, the chance that it fails at least once is 1 − 0.910, about 65%. Used twenty times, about 88%. If one failure is catastrophic, a 90% success rate is not safe at all.
Positive expected value is not enough
Worked example: A favourable bet that still loses money
Imagine you could repeatedly bet your whole savings on a coin toss: heads, your money rises 50%; tails, it falls 40%.
| Expected change per toss: 0.5 × 50% + 0.5 × (−40%) | +5% |
|---|---|
| One heads and one tails, in either order: 1.5 × 0.6 | 0.90 |
| After 10 tosses with 5 heads and 5 tails: 0.90^5 | 0.59 |
Each toss has a positive expected value, yet the most typical result after ten tosses, five heads and five tails, leaves you with 59% of your money. The average is pulled up by a few very lucky sequences, while most people lose. This is the same arithmetic as in compounding and CAGR: losses hurt more than equal gains help, and repeated bets compound.
The lesson for investors is that the size of possible losses matters as much as the average. Rare, large losses are called tail risk. A strategy that earns small gains most of the time and occasionally loses a lot, such as selling insurance on unlikely events, can look excellent for years before a single bad event wipes it out.
Good decisions and good outcomes
Because outcomes involve chance, a good decision can turn out badly and a bad decision can turn out well. Someone who puts all their savings into one tip and doubles their money made a poor decision that happened to pay off; someone who spread their savings sensibly and lost 10% in a bad year may have decided well.
Judging decisions only by results teaches the wrong lessons. A better habit is to write down, before deciding, what you expect, why, and what would make you wrong, then review the reasoning later as well as the result.
Why it matters
Investment claims often hide the probabilities: "this strategy wins 80% of the time" says nothing about how much is lost the other 20% of the time. Thinking in expected values, and asking about the worst case as well as the average, protects you from offers that sound safe but are not, and from drawing conclusions from one lucky or unlucky result.
Limitations
- In markets, probabilities are rarely known. They are estimates, usually based on past data that may not describe the future.
- Expected value describes the average over many repeats. Many real decisions happen only once.
- The "at least once" formula assumes each try is independent. In markets, bad outcomes often arrive together, which makes the risk worse.
Common mistakes
- The gambler's fallacy. After five heads in a row, tails is not "due". Each toss of a fair coin is still 50-50.
- Looking at the chance of winning but not the size of the loss.
- Treating a probability above 90% as a certainty, especially when the bet is repeated.
- Judging a decision only by its outcome.
Try it yourself
- Work out the expected value of a ₹500 bet that has a 30% chance of returning ₹1,500 and otherwise returns nothing.
- A trading strategy claims to work 95% of the time. If you use it 20 times, what is the chance it fails at least once? (Hint: 1 − 0.9520.)
- Think of a decision you made that turned out well. Would you make it again with the same information, knowing it could have gone the other way?
Key takeaways
- A probability measures how likely something is, from 0 (impossible) to 1 (certain). Over many repeats, it is roughly how often the event happens.
- Expected value = the sum of each outcome multiplied by its probability. It is an average over many repeats, not what will happen to you.
- A bet with a positive expected value can still ruin you if a single loss is large enough, so the size of possible losses matters as much as the average.
- Something with a 90% chance of working, repeated ten times, fails at least once about 65% of the time.
- Judge decisions by the information and reasoning behind them, not only by how they turned out.
Check your understanding
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Terms used in this lesson
- Probability
- A number from 0 to 1 describing how likely something is. Over many repeats, it is roughly how often the event happens.
- Expected value
- The average result of a decision if it could be repeated many times, found by weighting each outcome by its probability.
- Expected return
- The average return an investment would deliver if the same situation could be repeated many times, weighting each outcome by its chance.
- Tail risk
- The risk of rare but very large losses, which averages and everyday experience tend to understate.
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This lesson is general education, not investment advice. Examples use made-up or clearly dated figures to explain ideas; they are not recommendations. Read the disclaimer.