# Probability of profit versus expected value

**ADV-28 · Advanced Options · Core**

**Lesson question:** Can a high win rate hide a losing trade?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain the mechanism in the lesson question.
- Reproduce the worked comparison and recognize its failure case.

**Prerequisites:** OPT-F01, OPT-F02, OPT-F03, OPT-F04

Probability of profit measures frequency under a model, not profit size or loss severity. A high win rate can coexist with negative expected value. A low premium credit can finance a frequent gain while leaving rare large losses.

Model probabilities themselves also carry uncertainty.

## Understand the mechanism

Expected value weights every outcome by probability and subtracts costs. Tail losses, gaps and changing dependence can be missed by a limited backtest. Margin and funding can force a position closed before an assumed terminal outcome.

Do not convert option Delta into a universal probability of a profitable multi-leg strategy.

## Walk through the process

Report win rate alongside average win, average loss, distribution tails, costs and drawdown. Stress adverse assumptions and selection bias. Evaluate enough independent evidence to avoid extrapolating a lucky streak.

A successful recent period is not proof of stationary probabilities.

## Model and implementation boundary

These are complete fictional reading examples, not live opportunities. Terminal identities assume the stated common settlement. Multi-expiry values require a surviving-leg model. Costs, assignment and financing can alter realized outcomes; a mathematical bound alone does not guarantee the same operational loss path.

## A worked example

Suppose a fictional strategy wins 50 with 90% probability and loses 600 with 10%. Gross expected value is .9×50−.1×600=−15. With five cost per trade it becomes−20.

The graph varies assumed win probability, holding payoffs and costs fixed. It makes clear how much accuracy is needed merely to break even; it does not estimate any real strategy's probability.

## Calculation popup: Expected value

```text
EV=p×win−(1−p)×loss−cost
```

These probabilities are hypothetical, not estimates from market data.

1. .90×50=45.

2. .10×600=60.

3. 45−60−5=−20.

**Result:** Net expected value −20 per trade.

## Graph: Probability of profit versus expected value—worked comparison

**Horizontal:** Assumed win probability  
**Vertical:** Net expected value (USD per trade)

Fictional data from the written example. The chart organizes the comparison; its shape is not a forecast.

Complete numerical data are embedded in `scripts/advanced-options/ADV-28.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

Calling a short-premium trade safe because most historical outcomes won ignores rare severity and implementation costs.

## Check your understanding

### What is net expected value in the example?

**Answer:** −20 per trade.

**Why:** 45−60−5.

### What is the critical failure to avoid?

**Answer:** Calling a short-premium trade safe because most historical outcomes won ignores rare severity and implementation costs.

**Why:** Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome. **Critical risk check.**

## Takeaway

Return to the question: Can a high win rate hide a losing trade? State the observed facts, the assumptions and the failure condition separately.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [OIC Strategy Reference](https://www.optionseducation.org/strategies/all-strategies-en)

- [OCC Options Disclosure Document](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document)

- [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks)