Probability, expectancy and losing streaks

A win rate is only one ingredient

Winning often and making money are different claims. A process can produce many small gains interrupted by a few large losses. Another can lose frequently but occasionally gain more than its accumulated losses. To understand either, record the frequency and magnitude of outcomes, not just the number of green entries in a journal.

For a simplified two-outcome model, expected profit per trade equals win probability times average win, minus loss probability times average loss, minus average costs. This is a mathematical average under assumed probabilities. It is not a promise about the next trade, a particular month or the survival of an account. The probabilities themselves usually have to be estimated.

Calculate the net version

Suppose a hypothetical process wins 40% of the time, with a $180 average gross win and a $100 average gross loss. Its gross expectancy is 0.40 × $180 minus 0.60 × $100, or $12. If average round-trip costs are $15, net expectancy is negative $3. An attractive reward-to-risk ratio does not rescue a process from its actual costs.

Now consider a process that wins 80% of the time, gaining $20 on each winner but losing $100 on each loser. Before costs its expectation is $16 minus $20, or negative $4. Calling it an 80% accurate system describes its frequency but hides its economics. Rare losses also may be larger than anything seen in a short sample.

Respect uncertainty in the sample

A sample win rate is an estimate, not an intrinsic property engraved into the market. Ten wins in twenty observations provide limited evidence about the long-run distribution. Market conditions, execution quality and selection rules can change. Reporting the sample size and the observation period is therefore part of reporting the statistic itself.

The simple formula also compresses all winners and losers into averages. Inspect the full outcome list: a handful of extraordinary trades may account for the entire result. Recalculate without the largest winner, with higher costs, and under worse execution assumptions. These are sensitivity exercises, not predictions, but they reveal how dependent a conclusion is on favorable details.

There is a second uncertainty hidden inside a reported average: how variable the individual outcomes are. If two samples both average $10 per trade, but one swings between small gains and losses while the other contains a single $5,000 winner, they do not offer equally precise evidence about future expectancy. Under independent, identically distributed observations with finite variance, the standard error of the sample mean is the sample standard deviation divided by the square root of the sample size. Overlapping positions, clustering and changing conditions weaken that simple calculation.

Plan for sequences, not just averages

Even if independent trades each had a known 50% loss probability, four specified consecutive trades would all lose with probability 0.5 to the fourth power, or 6.25%. That is not the probability of seeing at least one four-loss streak somewhere in a long series; there are many possible starting points. Real outcomes may also cluster rather than remain independent.

A positive average does not prevent deep drawdowns, margin calls or abandonment of the process. A paper report should show the sequence of outcomes, cumulative equity and the worst decline from a prior peak. Arithmetic is useful because it exposes assumptions. It becomes dangerous when estimated inputs are presented with the confidence of known physical constants.

Keep units consistent when comparing strategies. A dollar expectancy from one position size is not directly comparable with another calculated at ten times the exposure. Expressing outcomes relative to an explicitly defined initial planned loss can help comparison, but gaps can produce losses greater than one such unit. Report actual cash outcomes alongside any normalized measure so the transformation does not hide their scale.

Turn an estimate into a range of plausible outcomes

Imagine observing 55 winners in 100 independent trials with an unchanged win probability. The observed win rate is 55%, but a 95% Wilson confidence interval is approximately 45.2% to 64.4%. That range is wide enough to include an unfavorable even-money process before costs. The confidence level describes the long-run coverage of the interval-producing procedure under its assumptions; it does not mean this particular interval has a 95% probability of containing a changing market parameter. NIST explains the Wilson method in its guidance on binomial confidence intervals.

For the earlier $150 winner, $100 loser and $10 cost example, net expectancy can be written as 250p − 110 dollars. At p = 0.40 it is −$10; at p = 0.50 it is $15. Holding payoff sizes fixed makes the effect of probability uncertainty visible. In a real dataset, average wins, average losses and costs are uncertain too. Moving only the win rate while treating the other estimates as exact understates the number of things that could change.

Compounding introduces a separate distinction. A 20% gain followed by a 20% loss leaves $1,000 at $960, even though the arithmetic average of those two percentage returns is zero. The correct wealth calculation is $1,000 × 1.20 × 0.80. This is why positive average trade returns cannot, by themselves, establish a desirable account-growth path. Exposure, reinvestment and the possibility of exhausting available funds matter.

A useful paper exercise therefore has three outputs: an estimate of net expectancy, an account path under stated sizing rules, and a sensitivity range for uncertain inputs. Keep any simulated range conditional on the model that generated it. Shuffling individual trades can illustrate sequencing, but it destroys dependence between neighboring outcomes; resampling blocks can preserve some local clustering without recreating shocks that never appeared in the source data.