Probability describes how likely a trading outcome is. Expected value combines that probability with the money gained or lost. Together, they answer a more useful question than “Will this trade win?”: does the estimated reward justify the risk and cost of taking it?
A strategy can lose more trades than it wins and still have positive expected value. Another can win almost every day yet lose money when an occasional large loss arrives. The difference lies in the size and frequency of outcomes, not the appeal of the win rate.
The calculation is straightforward. Establishing whether its inputs deserve confidence is the harder job.
What Probability Measures in Trading
A trading probability must refer to a clearly defined event. “The market will rise” leaves too much unanswered. Rise by how much, over what period, and from which entry price?
A usable definition might be: after a stated entry signal, the price reaches a profit target before a stop within the next trading session. Another might measure whether a position produces a positive return after five days, including transaction costs. These are different questions and can produce different probabilities from the same price history.
Define the outcome rules before examining results. Include positions that expire, close unchanged, or exit early. Otherwise, the analysis can exclude inconvenient trades without making that omission obvious.
Probability also differs from certainty about an individual trade. A model assigning a 60% chance of success still assigns a 40% chance of failure. A losing outcome does not, by itself, disprove the model. Equally, one winning trade does not validate it.
Expected Value Connects Probability With Payoff
Expected value is the probability weighted average of possible outcomes. Multiply each outcome by its probability, then add the results. This is the mathematical definition developed in MIT’s lecture on expectations of discrete random variables.
For a strategy with only winning and losing outcomes:
Expected value per trade = p × W − (1 − p) × L − C
Here, p is the probability of winning, W is the average gross profit conditional on winning, L is the average gross loss conditional on losing, expressed positively, and C is the average trading cost per completed trade.
Consider this hypothetical strategy, traded at a fixed size:
| Input | Assumed value |
|---|---|
| Probability of winning | 45% |
| Average gross winning trade | $180 |
| Average gross losing trade | $100 |
| Average cost per completed trade | $8 |
The expected value is:
0.45 × $180 − 0.55 × $100 − $8 = $18 per trade
The strategy has positive expected value despite losing 55% of its trades under these assumptions. Across 100 trades with the same expected dollar outcome, expected total profit is $1,800. That is an average across possible results, not a promised account balance.
Actual profits will vary around that expectation. The detailed guide to calculating a trading strategy’s expected value develops the calculation for different payoff structures and risk units.
Why Win Rate Alone Can Mislead
Suppose another strategy wins 90% of the time, earning $10 per winner, but loses $100 on each losing trade. Its expected value before costs is:
0.90 × $10 − 0.10 × $100 = −$1 per trade
Nine small wins do not compensate for one large loss. A high win rate can look reassuring while concealing poor economics.
The reverse also applies. A low win rate is not automatically a problem if winning trades are sufficiently large. Neither measure should be judged without the other.
Calculate Expectancy After Costs
Costs belong inside the calculation, not in a footnote. For the earlier example, increasing average cost from $8 to $20 reduces expected value from $18 to $6. At $26, the estimated edge disappears.
Use costs that match the trade’s holding period and execution method. These may include commissions, the bid and ask spread, slippage, financing, or borrowing charges where applicable.
Avoid counting the same cost twice. If recorded profits already reflect actual entry and exit fills, the effect of the spread and execution slippage may already be included. If commissions have also been deducted, subtracting them again understates performance.
Likewise, a planned profit target is not evidence of the average realized win. Partial exits, early closures, and imperfect fills can change the payoff. Estimate expectancy from outcomes the rules actually produce, not the outcomes written in the trading plan.
Estimate Probabilities Without Treating Them as Facts
The simplest estimate of winning probability is winning trades divided by total trades. If a strategy wins 55 of 100 trades, its observed win rate is 55%.
That figure is an estimate of an unknown probability, not a measurement with perfect precision. Under an independent, constant probability model, a 95% Wilson confidence interval for 55 wins in 100 trials is approximately 45.2% to 64.4%, using the NIST method for confidence intervals for proportions.
The interval illustrates how much uncertainty a modest sample can contain. Its 95% label describes the coverage of the procedure over repeated samples under the model. It does not mean there is a 95% probability that the next trade will win.
Winning probability is only one uncertain input. Average wins and losses are estimates too. A few unusually profitable trades can pull the sample average upward, while a short record may contain no example of a rare, severe loss.
Keep records of trade dates, signals, position sizes, realized outcomes, and costs. Expressing results relative to each trade’s initial planned risk can help separate strategy performance from changes in dollar size. Retain the dollar results as well, because the account pays its bills in money, not normalized units.
A Good Backtest Can Still Be a Poor Estimate
Choosing a strategy after testing many alternatives changes how its historical performance should be interpreted. The best result may reflect favorable noise as well as a genuine effect. This selection problem is examined in the research paper The Probability of Backtest Overfitting.
Freeze the trading rules before evaluating them on reserved data. Keep a record of rejected versions rather than presenting the eventual winner as the only idea tested.
Also check whether the test could have used each piece of information at the time of entry. A calculation based on future knowledge does not become tradable because the spreadsheet runs without errors.
Positive Expectancy Does Not Prevent Losing Streaks
Even a favorable probability model permits uncomfortable sequences. With independent trades and a constant 45% winning probability, the probability that the next five trades all lose is:
0.555 ≈ 5.03%
This describes one designated block of five trades. It is not the probability of encountering at least one five trade losing streak somewhere within a much longer record. That question includes many possible starting points.
The NIST binomial distribution reference provides the model for counting wins across a fixed number of trials with constant success probability. With independence assumed, it can describe the range of possible win counts rather than just their average.
Apply these assumptions cautiously. If several positions express the same market view, treating their outcomes as independent may understate how easily losses can arrive together.
A streak is therefore a reason to compare results with the model, not automatically abandon the strategy or increase its size. Under independence, previous losses do not make the next win due. Markets do not keep a fairness ledger.
Use Conditional Probabilities and Update With Evidence
A single average probability can hide differences between trading conditions. Suppose a hypothetical strategy wins 60% of trades in one predefined condition and 40% in another. Its combined win rate depends on how frequently each condition occurs.
The relevant question becomes: what is the probability of success given the conditions observable before entry? This is conditional probability.
Be careful with the word “observable.” Classifying a period as a trend after seeing the full price move does not establish that the same classification was available when the trade opened.
More detailed categories also leave fewer observations in each group. Splitting a small dataset by instrument, session, volatility, and signal type can create precise looking percentages supported by very little evidence.
Bayesian analysis of trading probabilities provides a framework for combining prior assumptions with new observations. The practical principle is to update estimates in proportion to the strength of the evidence, rather than swinging between certainty and panic after each result.
Set review rules in advance. Otherwise, changing assumptions after every loss can make a supposedly systematic process depend on the latest emotional reaction.
Expected Value Does Not Describe the Whole Risk Distribution
Two strategies can share the same expected value while presenting very different risks. One might produce modest gains and losses. Another might produce frequent small profits interrupted by rare, much larger losses.
The mean does not show how widely outcomes vary, how severe the worst outcomes might be, or whether losses tend to cluster. Those questions require examination of the return distribution and the sequence of results.
Financial returns can exhibit heavier tails than a normal distribution and periods of clustered volatility. These empirical properties are documented in Rama Cont’s research on asset return distributions. A convenient bell curve should not be treated as proof that extreme outcomes are negligible.
There is also a distinction between an asset’s return distribution and a strategy’s profit distribution. Entry rules, exits, holding periods, and position sizes change how price movements become trading outcomes.
Stress the payoff assumptions rather than only the winning probability. What happens if the average loss rises, costs increase, or several positions lose together? What if the strongest historical winners do not recur?
A strategy whose estimated expectancy turns negative after a small, plausible adjustment deserves more caution than its headline average suggests. The guide to normal distributions, fat tails, and extreme market moves examines these distribution assumptions in more detail.
Position Size Determines Whether the Risk Is Bearable
Expected value does not tell you how much capital to commit. Under a simplified model with proportional payoffs and unchanged costs, doubling position size doubles both expected dollar profit and the size of losses.
That does not make doubling the position a better decision. A strategy can have positive estimated expectancy while exposing the account to losses it cannot withstand.
Percentage returns also compound rather than add. Starting with $100, a 50% gain followed by a 40% loss leaves $90. The arithmetic average of those two returns is positive, but the account has lost money.
Use position sizing and capital allocation to connect an uncertain edge with an acceptable exposure. Account for positions already open, not just the proposed trade in isolation.
Smaller sizing cannot turn a genuinely negative expected value strategy positive. It can, however, reduce the damage while an uncertain estimate is being evaluated.
Make the Trading Decision From the Full Model
A practical review should answer four questions:
- What outcomes are being measured? Define entries, exits, holding periods, and treatment of every completed trade.
- Is expected value positive after costs? Use realized or defensible payoff estimates, not targets alone.
- How uncertain are the inputs? Examine sample size, strategy selection, and changes in trading conditions.
- Can the account tolerate unfavorable outcomes? Consider loss size, simultaneous exposures, and adverse sequences.
These questions separate a probability estimate from a trading decision. A positive number in an expectancy calculation is a starting point for evaluation, not permission to ignore risk.
The objective is not to predict every trade correctly. It is to identify a defensible advantage, allow for the possibility that the estimate is wrong, and keep exposure consistent with that uncertainty.