Chapter 8 · Expected Value
Is a trade with only a 40% chance of winning worth considering?
- Skills to practice
- Probabilistic decisions
- Read first
- Chapter 7 · Probability
- 3D simulation
- None
Market scene
Zhe studies a teaching rule: 40% chance to win $2,000 and 60% to lose $1,000. No fees yet, with fixed amounts.
Lin asks why research something losing more often than winning. Zhe multiplies each outcome's dollars by its chance. He does not immediately trade; he checks omitted outcomes first.
Why might a mostly losing rule still deserve research?
Your decision
With only those assumptions, what will you do?
Observe the result
| Outcome | Chance | Trade PnL | Mean contribution |
|---|---|---|---|
| Target reached | 40% | +$2,000 | +$800 |
| Exit price reached | 60% | −$1,000 | −$600 |
| Total | 100% | No third single outcome | +$200 |
An average may never occur in a single trade. One winner cannot establish accurate probabilities.
The mechanism
Two outcomes have 40% and 60% chances, winning 2,000 or losing 1,000. Exclude compounding/costs initially. Repeated-rule averages weight every outcome, not merely the more frequent side.
- List all outcomes
- Specify their probabilities
- Multiply by PnL amounts
- Sum contributions
- Deduct per-trade costs
- Check inputs and tolerance
Use planned loss as one unit: wins are two, losses one, average +0.2. This compares account sizes. Extra costs of 0.25 units make the average −0.05.
Real trades include unfilled orders, gaps, or early exits, not forced binary outcomes. New scenarios require reallocating probabilities to remain exhaustive.
What it is called
Real markets
Chapter 14 compares execution across depths. Screen-price target returns without actual depth costs overstate net results.
Chapter 15 includes spreads, inventory, and adverse selection. Spread-winning fills alone omit counterparty-related losses.
Chapter 17 separates price PnL and funding. Compare expressions with holding cash flows in scenarios, not direction alone.
Hands-on
Assume independent outcomes. A win earns the reward-to-risk amount in R; a loss costs 1R. The same cost is deducted on each trade. R is the stop-loss amount planned in advance.
Seed1; changing payoffs or costs leaves this outcome sequence unchanged. A sample win rate does not promise the next outcome, and the same sample cannot validate the probability you entered.
Course versionV1-docs; sourcelab:probability；Chapter 8 / TRD-PROB-002
Records parameters and results at the click only; does not mean the experiment passed.View snapshot to save
- Keep 40% wins, payoff 2, costs 0. Record the mean and actual sample sum.
- Change costs only to 0.25R and explain the crossing of zero.
- Generate another group and inspect whether sample totals always share the model mean's sign.
- Write whether research continues, covering evidence, omissions, cost definitions, and risks. No registration or orders.
Change one variable
Three depths
- FoundationIs a trade with only a 40% chance of winning worthwhile?Chapter 8
- AdvancedHow can historical samples estimate expectancy, and how large is estimation error?Advanced A · Probability and statistics
- InstitutionalWhat expectancy remains after costs, capacity constraints, and tied-up capital?Institutional
List exhaustive outcomes before calculating. A positive number is only research's first step; inspect probabilities and worst scenarios.
Historical estimates need fixed entries, exits, costs, and full trade distributions. Express errors with resampling/intervals; dependent returns cannot blindly sample independently. See Advanced A.
Continue the project: estimate net expectancy and uncertainty.
Institutions compare net returns with costs, funding use, and capacity. Larger impact changes distributions; small-order averages cannot scale infinitely. Risk teams inspect tails and exits.
Continue the project: include costs and capital use in efficiency.
Questions to take away
Chapter self-test
800 − 600 = +$200, or 0.2R of planned $1,000 risk.
It is a weighted average. Single outcomes remain +2,000 or −1,000.
No. Negative-expectancy rules also have profitable scenarios; inspect distributions and evidence.
Deduct from both outcomes; average becomes 200 − 250 = −$50.
One idea to take away
Expected value, rather than win rate alone, determines whether a trade is worthwhile: sum each outcome's probability multiplied by its PnL.
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Is a trade with only a 40% chance of winning worthwhile?
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