Reaching a funded account through a two-step challenge is not simply the product of the two phase pass rates. Multiplying the two correct phase probabilities gives a lower bound, not the answer: it is exact only for a trader whose edge is a known constant, and because a real win rate is an estimate and both phase probabilities rise with it, the composed probability is at least the product — a sweep found no exception across 6,144 combinations. On this article's illustrative settings — 0.5% risk per trade, reward-to-risk 2, phase one 8% target against a 6% floor, phase two 4% from a fresh start, 40% win rate estimated from 60 trades — phase one clears 78.1% and the shorter phase two 83.1%; their product, 64.9%, rises to a composed 70.5% once the shared-edge conditioning is added (+5.6 points), while the common shortcut of squaring one number understates further, at 61.0%. A second phase still costs something (70.5% is below phase one's 78.1%), and the largest single effect is the reset — two full journeys from the same balance, not half a journey each: one 12% run against the same floor reaches 75.9%, so resetting costs 5.4 points here and 5.9 at its worst near a 36% win rate. Free path: /app?src=article-two-step-challenge-probability.
Your numbers next · Working backwards: the 64.9% product (78.1% × 83.1%) is a floor — conditioning lifts it to 70.5% (+5.6), and the reset costs 5.4 you cannot multiply away.
Run the free personal rule editor against the same two-phase composition — no signup; computed in-browser.
Run the free path — the 64.9% product is a floor; find your composed 70.5% →
Is being funded really the product of the two phase probabilities?
Yes — on one condition, and it is a condition nobody meets.
Take a two-phase evaluation with the numbers set out beside this clause. Phase one asks for 8% of the starting balance in profit and ends the account if the balance falls 6% below where it began. Phase two asks for 4% against the same allowance, and it begins again from the starting balance: fresh equity, fresh high-water mark, fresh floor. The trader risks 0.5% of the starting balance per trade and a win pays twice what a loss costs, so the account walks a lattice and each phase is a race between a target above and a floor below.
Now hold the trader's per-trade win rate fixed and call it known. Phase two shares nothing with phase one — not the equity, not the peak, not the remaining room — because the reset deletes all three. Conditional on that win rate the two phases are independent, and the probability of clearing both really is the first probability multiplied by the second. Exactly, with no approximation.
The reset is what buys that. An allowance that carried across the join would tie the phases together through the room phase one had already spent, and the product would then be wrong in the opposite direction from everything below.
Multiplying is not an approximation to the answer. It is the answer, for a trader whose edge is a known number.
No such trader exists. What a trader has is a record, and a record yields an estimate. An estimate is not a number; it is a distribution over numbers, and both phase probabilities are functions of the same one.
The worked example is a hypothetical trade profile, not a measurement of any strategy and not a transcription of any firm's rules. Its numbers come from figures.data.js, which ships beside this page and prints what it emits when run directly.
Risk per trade 0.5% of the starting balance · reward to risk 2 · phase one 8% against 6% · phase two 4% against 6% · win rate 40% estimated from 60 trades · floor static · no time limit.
Nothing here is sampled. Phase probabilities solve in closed form and the estimate is integrated over, not simulated. The one place a seed appears is the trade-by-trade check in gates.js, which runs at seed 7.
Which way is the product wrong, and can it ever be wrong the other way?
It understates. It cannot overstate. That is not a finding about these particular numbers; it follows from two facts that hold for every setting of every control on this page.
First, both phase probabilities rise together. Couple two win rates by giving them the same stream of random draws, so that the higher win rate wins every trade the lower one wins and some the lower one loses. Then the higher account's balance is at or above the lower one's after every trade, and the gap between them never shrinks. It therefore reaches any target no later, and — because its running peak can only have risen by at least as much as its balance — its distance below its own peak is never greater. It clears the phase whenever the lower one does. This argument does not care whether the floor is a fixed line or one that trails a high-water mark, and it applies to both phases.
Second, two quantities that rise together cannot average apart. Take two independent copies of the unknown win rate. Whichever is larger, both phase probabilities are at least as large for it, so the product of the two differences is never negative. Its average is therefore never negative either, and expanding that average gives exactly twice the difference between the mean of the product and the product of the means. So the mean of the product is the larger.
Those two facts together say that the probability of clearing both phases is at least the product of the two phase probabilities, with equality only when the win rate carries no uncertainty at all.
The inequality above is arithmetic, not simulation, and it is reproducible by hand for any two quantities that move together. The version stated here is the standard association inequality for similarly ordered sequences.
Its only assumption is the monotone coupling in the paragraph above it. Where a rule breaks that monotonicity — a clause that penalises a trader for winning too fast, for instance — the sign is open again.
| 64.9% | the product of the two phase probabilities |
| 70.5% | the probability of clearing both |
| 5.6 | points the product leaves out, at 60 trades of record |
The product of two phase probabilities is a lower bound on the chance of being funded. It is never a ceiling, and how far below it sits is set by how little the trader knows about their own edge.
At the worked settings the product returns 64.9% and the composition returns 70.5% — 5.6 points, or 7.9% of the answer, that multiplying does not count. Sweeping 6,144 combinations of win rate, record length, targets and floors, the smallest difference found between the composition and the product is zero to the resolution of the arithmetic. Not one combination reverses the sign.
The useful way to say this is not that the formula is wrong. The formula is fine. The second factor is wrong: what belongs there is not the phase-two probability from the catalogue but the phase-two probability of a trader who has already cleared phase one, and those are different numbers.
What does clearing phase one actually tell you?
Less than the headline suggests about how good the trader is, and much more about how bad they are not.
Everything here is written for one trader who does not know their own edge. It reads identically for a cohort: if buyers differ in skill, the share of buyers who reach funding exceeds the product of the two phase pass rates by the same covariance, and a firm quoting per-phase rates would understate its own funding rate.
Which reading applies depends on whose uncertainty is being described, and the arithmetic does not distinguish them.
| 50.3% | chance the true win rate is below the estimate |
| 38.2% | the same chance, once phase one is cleared |
| 7.2 | points that adds to the phase-two probability |
Before phase one, the record leaves a 50.3% chance that the true win rate is below the estimate — which is roughly what an estimate means. After phase one has been cleared, that chance is 38.2%. The centre of the distribution barely moves: its mean goes from 40.2% to 42.0%. Almost the entire change is the left tail being deleted, because the traders in that tail are exactly the ones who did not get through.
Selection does not promote the middle. It removes the bottom — and the bottom is where the phase-two failures were going to come from.
Carried into phase two, that reweighting lifts the probability from 83.1% to 90.3%: 7.2 points bought by nothing except the information that the account is still alive.
The two answers, side by side
Set an edge, a record length, and each phase’s target and floor. The instrument returns both phase probabilities, the product of them, and the probability of clearing both — computed by carrying one trader through both phases rather than by multiplying.
What the record shows. A win pays twice what a loss costs.
%How much history that estimate rests on. Drag it and watch the last readout.
The instrument computes the composition by carrying one trader through both phases — the same unknown win rate on both sides of the reset — and the product by multiplying the two averages, so the difference between the readouts is the conditioning and nothing else. Every phase probability behind them is solved as a linear system across the lattice between floor and target. On each build those numbers are checked against model.js, which reaches them through the closed form of the recurrence instead, and against a trade-by-trade run that simulates the whole thing the slow way and agrees to within its own sampling error.
Which of the three suspicions about the toll is the biggest?
Three things about a two-phase evaluation make the ordinary account of it suspect, and they do not point the same way.
The second target is shorter. Illustrations that quote one probability and square it treat both phases as equally hard. Against the same floor, a shorter target is a shorter race: the phase-two probability here is 83.1% against phase one's 78.1%. Correcting for that alone moves the answer from 61.0% to 64.9% — worth 3.9 points, upward.
The phases are not independent draws. That is the conditioning above: another 5.6 points, also upward.
The account is reset between them. This one is not a correction to the arithmetic at all — the product already prices it, because each phase probability is computed from the starting balance. It is a correction to the picture, and it is the one that costs.
A two-phase evaluation is not one journey to a combined target with a checkpoint in the middle. It is two journeys, each starting at the same balance, each with the whole allowance to survive again. Profit banked in phase one buys no room in phase two. Measure it by running the same total requirement of 12% once, against the same floor, instead of twice.
The single run to the combined target is a measurement device, not a product anyone sells. It exists here to isolate one variable — whether the balance and the floor restart — while the target, the allowance, the risk and the edge are all held fixed.
| 75.9% | one run to the combined target, one floor |
| 70.5% | the same target as two reset phases |
| 5.9 | points at the worst win rate, 36% |
One run reaches 75.9%. The two reset phases reach 70.5%. The reset costs 5.4 points at these settings and 5.9 points at its worst, around a win rate of 36% — where the outcome is genuinely in doubt and a deleted buffer decides it. Far from that region the cost fades, because a trader who was going to clear the target comfortably clears it either way and a trader who was not fails either way.
The largest of the three effects is the one the formula never claimed to describe. Two phases are not half a journey each. They are two full ones, and only the target was halved — the floor has to be survived twice over.
Between the two corrections that are about the formula, which dominates depends on the record. Below 98 trades of history the conditioning is the larger; above it the shorter second target is, and the conditioning fades toward nothing while the target effect does not fade at all.
Which floors and clauses does this composition not cover?
The floor modelled here is static: a fixed line below the starting balance. The monotone-coupling argument in the second clause covers a floor that follows the account's high-water mark upward as well, so the direction of the correction is safe there, but none of the sizes on this page were computed for one.
Only two clauses are modelled — a profit target and a maximum drawdown. A daily loss limit, a minimum number of trading days, an inactivity rule, a consistency cap or a time limit would each change both phase probabilities, and the second and third of those can bind across the join in ways this composition does not see. A rule that couples the phases directly breaks the conditional independence that makes the product exact in the first place.
There is a larger effect than either correction discussed here, and it is a different question: the gap between a pass probability computed at a point estimate and one averaged over what the record actually supports. At the worked settings that difference is bigger than the conditioning and the shorter target combined. This page holds it out of the comparison deliberately — both methods here are fed the same, correctly averaged, per-phase numbers, so that the only thing being compared is how they are combined.
The uncertainty modelled is uncertainty about a win rate, given a record of independent trades with a fixed reward-to-risk ratio. Real records are not independent, the reward-to-risk ratio moves, and a trader's edge is not constant across the weeks a two-phase evaluation takes. Each of those widens the distribution the argument integrates over, which makes the correction larger rather than smaller — but none of them is measured here.
No firm's rules are transcribed on this page and no pass rate is attributed to anyone. The shape modelled — a shorter second target, a reset balance and a reset floor — is set by the reader, not asserted as an industry fact.
The common shorthand for a two-step evaluation treats the two phases as independent and multiplies their probabilities — a law of composition: probabilities multiply, so a second phase always costs something. Turned into money, that gives the expected number of attempts as one divided by the product, and the real cost as the fee divided by it. Both follow immediately once the multiplication is treated as exact.
The second half of the composition claim survives: a second gate always costs something, because the post-selection phase-two probability is still below one. The first half does not. Multiplying is a special case, valid when the two probabilities are known constants, and a lower bound whenever they are estimates. The published number is conservative, and it is conservative by a margin that depends on the reader's record — which makes the attempts and the cost divided out of it the high estimates rather than the low ones.
Where does multiplying still trip people up?
Can multiplying the two phase probabilities ever overstate the chance of being funded?
Not while both phase probabilities move the same way with the trader's edge, which they do whenever the rules are a profit target and a drawdown floor. Under that condition the composed probability is at least the product, and the sweep behind this page found no exception in 6,144 combinations of win rate, record length, targets and floors. A rule that made one phase easier for a worse trader would break the condition, and then the sign is open.
If I knew my win rate exactly, would multiplying be correct?
It would be exactly correct, because the reset between the phases removes every other channel through which the first phase could inform the second. That is the whole point: the error is not in the arithmetic of composition but in forgetting that both inputs are estimates of one shared unknown. Push the record length on the instrument above toward its maximum and the two answers converge.
Does a second phase always cost something?
Yes. Even after the conditioning correction, the phase-two probability of a trader who has already cleared phase one is below one, so the composed probability is below the phase-one probability alone. At the worked settings it falls from 78.1% to 70.5%. What is not true is that the toll is as large as multiplying two catalogue probabilities makes it look.
Is a shorter second target the same thing as an easier second phase?
Against an identical floor, yes, and that is why the phase-two probability here is the higher of the two. But it is easier only relative to the same fresh start. The shorter target does not compensate for the reset: running 8% and then 4%, each from the starting balance, is worse than running 12% once against the same floor, by 5.4 points at these settings.
Working backwards: the 64.9% product is a floor, not your answer
Multiplying the two phase pass rates (78.1% × 83.1% = 64.9%) is a lower bound; conditioning lifts the composed probability to 70.5% (+5.6), and the reset costs 5.4 points you cannot multiply away. Open the free personal rule editor before treating the catalogue product as the answer.
Open the free path — carry one trader through both phases: 64.9% product, 70.5% composed →
Every number in the prose and in the figures is emitted by figures.data.js, which ships beside this page; every number the instrument shows is computed in the page and checked on each build against model.js, an independent implementation, and against a trade-by-trade simulation at seed 7. The trade profile and the rule set are illustrative — a hypothetical account used to make the composition arithmetic concrete, not a measurement of any strategy and not a transcription of any firm's published rules.
- W. Feller, An Introduction to Probability Theory and Its Applications, Volume I — the ruin problem for a random walk between two absorbing barriers, which is the form each phase takes here. Read as background for the closed form; the derivation used on this page is given in figures.data.js.
- G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities — Chebyshev's inequality for similarly ordered sequences, the general form of the association argument in the second clause. The one-line proof is reproduced above so the claim does not rest on the citation.
- PropSurvival, Monte Carlo pass probability: the two errors, and which one you can fix — the treatment of estimation error itself, which this page holds out of scope and cites rather than repeats.