The maths of passing: risk per trade, trades needed, probability of passing and the expected cost of attempts

What you will learn

• How to estimate the number of trades an evaluation will take from your expectancy and risk per trade.
• A pocket formula for the probability of reaching the target before the loss limit.
• Why smaller risk per trade helps a trader with an edge and does nothing for a trader without one.
• How to turn the pass probability into an expected total cost of attempts.

Explanation

R-multiples, win rate and expectancy were covered in the course Risk & Trade Management. Here we apply them to a specific question: what is the chance of hitting a profit target T before hitting a loss limit L if every trade risks r percent of the account?

Step 1: trades needed

If your expectancy is E in R-multiples per trade (for example 0.04 R), the expected gain per trade is r × E percent of the account. Trades needed to reach the target are approximately T / (r × E). With T = 8 percent, r = 1 percent and E = 0.04 R: 8 / (1 × 0.04) = 200 trades. Halving r doubles the number of trades to 400. This is the first trade-off: lower risk means more trades, more time and more commissions.

Step 2: the pocket pass formula

Model each trade as a step of +1 R with probability p and −1 R with probability q = 1 − p (a 1:1 reward-to-risk system; other ratios can be converted approximately by using the win rate of an equivalent 1:1 system). Let a = T / r be the number of winning steps needed and b = L / r the number of losing steps that end the attempt. The classic gambler's-ruin result gives the probability of reaching +a before −b:

P = (1 − (q/p)^b) / (1 − (q/p)^(a+b)) for p ≠ 0.5, and P = b / (a + b) for p = 0.5.

The no-edge case is the most instructive: with T = 8 percent and L = 10 percent, a coin-flipping trader passes phase 1 with probability 10 / 18 = 55.6 percent, regardless of risk per trade. A 5 percent phase 2 with the same 10 percent limit gives 10 / 15 = 66.7 percent. Both phases: 0.556 × 0.667 = 37 percent, so roughly one in three traders with no edge passes by chance in this simplified model. That is why "I passed" proves little, and why lesson 7 looks at survivorship.

Step 3: expected attempts and cost

If each attempt passes with probability P, the expected number of attempts is 1 / P and the expected fee cost is fee / P, ignoring discounts, free retries and the time you spend.

Worked example

Hypothetical example: fee €500, phase 1 target 8 percent, phase 2 target 5 percent, static maximum loss 10 percent in each phase, 1:1 reward-to-risk, no daily loss limit, no costs. We compare three traders by win rate and two risk settings. All numbers are rounded.

Calculation for p = 0.52, r = 1 percent, phase 1: q/p = 0.48 / 0.52 = 0.923; a = 8, b = 10. P = (1 − 0.923^10) / (1 − 0.923^18) = (1 − 0.449) / (1 − 0.237) = 0.551 / 0.763 = 72.2 percent.

Read the table in three lines. With no edge, risk per trade changes nothing; luck decides. With a positive edge, smaller risk raises the pass probability because more trades let the edge show, at the price of more trades and more time (for p = 0.52, E = 0.04 R: 200 trades at 1 percent, 400 at 0.5 percent). With a negative edge, smaller risk makes failure more certain, and the expected cost explodes. The evaluation is a test of whether your edge survives many trades, and risk per trade only sets how long the test runs.

Real evaluations are harsher than this model: a daily loss limit cuts off streaks, spreads and commissions lower the effective win rate, and consistency rules add trades. Treat the model as an upper bound on your pass probability.

Common mistake

Practice activity

On paper: fee €300, one-phase evaluation with target 10 percent and static maximum loss 8 percent. Trader X has p = 0.50; trader Y has p = 0.55 (1:1 system). Both risk 1 percent per trade. Calculate for each the pass probability, the expected number of attempts and the expected fee cost. Use 0.45 / 0.55 = 0.818, 0.818^8 = 0.200 and 0.818^18 = 0.027.

Key takeaways

• Trades needed ≈ target / (risk per trade × expectancy); lower risk means more trades.
• The gambler's-ruin formula gives a quick upper bound on your pass probability.
• Without an edge, risk per trade does not change the odds; with an edge, smaller risk helps.
• Expected cost of attempts = fee / pass probability; calculate it before buying.

Next step

Passing is only the start; the next lesson compares what you actually receive afterwards: profit splits, first-payout rules, buffers, caps, refunds and scaling plans.