Probability of ruin with a daily limit
Lesson 11 · about 10 min
The previous lesson said that variance can fail a trader with a real edge. This lesson puts numbers on it, first with the simple gambler's-ruin model, then with a daily loss limit added, which is where the model stops being academic and starts explaining why oversized accounts die on a Tuesday.
The model
Assume each trade wins or loses exactly 1R (a 1:1 payoff) with win rate p. The account starts with D units of drawdown room and needs T units of profit. With no other rules, the probability of reaching T before losing D is:
P(pass) = (1 - (q/p)^D) / (1 - (q/p)^(T+D)), where q = 1 - p
(When p = 0.5 this reduces to D / (T + D) from the last lesson.)
Real trading has unequal wins and losses and a variable R, but the shape of the results holds, and the calculator at /tools/prop-firm-challenge lets you use your own win rate and payoff ratio.
Pass probability by win rate and size
Account: "$50,000", target $3,000, drawdown $2,000 (static for now).
| Win rate (1:1) | Risk $400 (T=7.5, D=5) | Risk $200 (T=15, D=10) | Risk $100 (T=30, D=20) | Risk $50 (T=60, D=40) |
|---|---|---|---|---|
| 45% | 15% | 4% | 0.2% | ~0% |
| 50% | 40% | 40% | 40% | 40% |
| 52% | 52% | 64% | 81% | 96% |
| 55% | 69% | 87% | 98% | ~100% |
| 60% | 87% | 98% | ~100% | ~100% |
Read the rows.
- No edge (50%): size does nothing. Every column is 40%.
- Negative edge (45%): bigger size helps. At $400 risk you have a 15% shot; at $50, none. A trader with no edge who wants a lottery ticket should size up, and the firm's funnel is full of exactly those traders.
- Positive edge (52% to 55%): smaller size helps, dramatically. A 52% trader goes from a coin-flip at $400 to 96% at $50. The reason is that small size gives the edge enough trades to overwhelm the noise before the noise can reach the drawdown line.
The 52% row is the important one. That is a thin, realistic edge, and at the right size it passes almost every time; at the wrong size it is a coin flip.
Key idea: With a real edge, the pass probability is a function of size. Cutting risk per trade from 1/5 to 1/10 of the daily limit takes a 52% trader from 64% to 81%, and a 55% trader from 87% to 98%. The cost is only time.
Now add the daily loss limit
Introduce a $1,000 daily limit and assume 3 trades per day. Simulating the same set-up (static drawdown, daily limit, 20,000 runs per cell):
| Win rate | Risk $400 | Risk $200 | Risk $100 |
|---|---|---|---|
| 45% | 7% | 4% | 0.2% |
| 50% | 18% | 40% | 40% |
| 52% | 25% | 64% | 81% |
| 55% | 36% | 87% | 98% |
| 60% | 55% | 98% | ~100% |
The $200 and $100 columns barely moved: at those sizes, three losses in a day is $600 or $300, well inside the $1,000 limit, so the daily rule almost never triggers. The $400 column collapsed. Three straight losses is $1,200, which breaches the daily limit before the drawdown allowance is even close. At 55% win rate the chance of three straight losses is 9% for any given run of three trades; over a 20-day evaluation that happens to nearly everyone.
The daily limit is what turns "too big" from a slower path into a dead one. It adds a second, much closer, failure line that only the oversized trader can reach.
Losing streaks are normal
Traders underestimate streaks. The longest losing streak you should expect in 100 trades is roughly ln(100) / ln(1 / loss rate):
| Win rate | Expected longest losing streak in 100 trades |
|---|---|
| 45% | about 8 |
| 50% | about 7 |
| 55% | about 6 |
| 60% | about 5 |
A 55% trader should expect a 6-loss streak somewhere in a 100-trade evaluation. At $400 risk that is $2,400, more than the whole drawdown allowance. At $200 it is $1,200, survivable if it spans two days. At $100 it is $600. Size so that the expected worst streak fits comfortably inside the allowance, not just the average day.
Two honest caveats
- These tables assume the drawdown is static. An intraday trailing drawdown shrinks D for most of the evaluation, lowering every number in the small-size columns slightly and the large-size columns more.
- The model assumes your win rate does not change under pressure. In practice it does, in the wrong direction, after losses. Smaller size also protects against that, because a $100 loss is easier to shrug off than a $400 one.
Try it: Open /tools/prop-firm-challenge, enter your own win rate and average win-to-loss ratio from your log, and run the target, drawdown and daily limit for the evaluation you are considering at three sizes. Write down the size at which your pass probability stops improving much; that is your maximum.
Recap
- With no edge, size does not change pass probability; with a negative edge, bigger size is a lottery ticket; with a positive edge, smaller size raises pass probability sharply.
- A 52% win-rate trader goes from a coin flip at $400 risk to over 80% at $100 on a $3,000 / $2,000 challenge.
- Adding a daily limit collapses the oversized column: three straight losses at 2/5 of the daily limit fails the day.
- Expect a losing streak of 6 to 8 in 100 trades; size so that streak fits inside the allowance.
See it drawn
Original diagrams for the ideas on this page. Illustrative, not real market data.