How to determine the moment of stopping by probability

Why you need a "moment of stopping by probability"

Stopping is a predetermined event in which you stop the game/session because the probability of an unfavorable outcome has exceeded the permissible threshold, or, conversely, the goal has been achieved. Unlike emotional "enough," probabilistic stopping relies on:

1. Outcome barriers (profit/drawdown);

2. Odds estimate (p, EV, variance);

3. Risk-metrics (Risk of Ruin, probability of false conclusion, confidence intervals);

4. Stop tests (SPRT/Bayesian rules).

1) Basic model: two absorbing barriers (target and stop)

Imagine capital varying in steps (rate/round): up with probability (p), down with probability (q = 1-p). We introduce two barriers: upper (+ T) (profit target) and lower (-L) (stop loss). As soon as capital reaches one of them - stop.

Probability of reaching the goal before the foot (class "player ruin")

If the steps are the same in absolute value and (p\ne q), then when starting at 0, with goals in steps (N = T/\Delta) up and (M = L/\Delta) down:

[
\ mathsf {P} {\text {get to} + T}; = ;\frac {1- (q/p) ^ {M}} {1- (q/p) ^ {M + N}}
]

При (p=q=0{.}5): (\mathsf{P}=\frac{M}{M+N}).

Rule: Select (T) and (L) so that (\mathsf {P}) matches your target success rate (for example, ≥ 60%). This is a stop on the barrier: we have reached one of the levels - to get out.

Practical conclusion: at unfavorable (p\le 0. 5) symmetrical goals and feet give ≤50% success. Only asymmetry of barriers (smaller stop, larger target) or actual (EV> 0) can be compensated.

2) Stop at risk of ruin (RoR) towards the end of the horizon

Let's say you have a bank (B), bet as a share (f), round volatility (\sigma), advantage (e) (expected return per round). For the final horizon (N), you are interested in: "What is the chance of falling below the critical level (B_{\min}) to the end?" If the conditional RoR at the current drawdown (DD) has become ≥ the specified threshold (\beta) (for example, 5%), we stop.

Working heuristic: if you play shares from Kelly, then when falling into the maximum permissible drawdown (for example, 20-30% with half-Kelly) - stop until the parameters are restored (recalculation (p, e ,\sigma), decrease (f)).

3) Confidence stop for win/probability

When the true odds (p) are unknown (slots, live markets), you update the observation score. Let there be (w) "successes" in binary abstraction for (n) attempts. Construct a two-sided 95% CI for (p) (e.g., Clopper-Pearson). If the CI upper bound for your real EV drops ≤ 0, the rule is:

💡 Stop, because with the current data, even a favorable assessment does not confirm a positive expectation with sufficient confidence.

Reverse: If the lower limit of the CI for (p) is above the threshold that makes your EV> 0, you can continue to the nearest profit/time barrier.

4) Bayesian stop: "probability that EV ≤ 0"

Set the prior to (p) (beta distribution (\text {Beta} (\alpha _ 0 ,\beta _ 0))). After (w) "successes" in (n) trials, the posterior (\text {Beta} (\alpha _ 0 + w ,\beta _ 0 + n-w)). Recalculate the posterior probability of the hypothesis "(EV\le 0)" (taking into account the payoff coefficients).

Rule: if (\mathsf {P} (EV\le 0\mid\text {data} )\ge\tau) (for example, 80-90%), - stop.

Pros: smooth accounting of a priori information, stability in small samples.

5) Wald sequential test (SPRT) - "online solution"

SPRT tests (H_0) vs. (H_1) on the fly, after each outcome. You set acceptable errors: (\alpha) (false alarm) and (\beta) (skip advantage), and two hypotheses by (p):

(H_0:; p = p _ 0) (boundary where EV ≤ 0), (H_1:; p = p _ 1) (expected benefit).

Log-likelihood ratio (LLR) is considered.

Stopping rules:

If LLR ≥ (\ln\frac {1-\beta} {\alpha}) → accept (H_1) (advantage confirmed) or exit on target.
If LLR ≤ (\ln\frac {\beta} {1-\alpha}) → accept (H_0) (no advantage) and stop.
Otherwise, continue to collect observations.

Where useful: when evaluating "live/dead" strategy in live or under new conditions of promo/coefs.

6) Three practical "stopping rules" (can be applied together)

1. Outcome barriers (T/L):

Fix the profit target (+ T) and stop loss (-L) in advance, consistent with the desired probability of success (\mathsf {P}) (formula in § 1). We reached one of the barriers - the way out.

2. EV confidence rule:

After each block of (k) rounds, recalculate the CI/Bayesian probability. If trust in EV> 0 is insufficient (CI includes 0 or (\mathsf {P} (EV\le 0 )\ge\tau)) - stop.

3. Ruin risk rule:

If the conditional RoR before the end of the horizon is ≥ (\beta) or the limit of permissible drawdown is reached (for example, 20% for half-Kelly) - stop, even if the goal is not reached.

7) Mini calculators (paper)

A. Match T/L to target success rate

Enter a score (p) (or range).

Select step (\Delta) and target (M = L/\Delta), (N = T/\Delta).

Calculate (\mathsf {P}) from the formula § 1. Match (M, N) to (\mathsf {P }\ge P_{\text{target}}) (for example, 60%).

Fix the barriers and do not change along the way (otherwise the mathematics of stopping breaks down).

B. Verification of EV confidence (frequency approach)

Every (k) rounds, construct 95% CIs for (p).

Recalculate EV taking into account payments and commissions.

If the upper (for a negative hypothesis) or lower (for a positive hypothesis) boundary of the DI intersects 0, stop/continue according to the rule § 3.

C. Bayesian trigger

Prior (\text {Beta} (1,1)) (neutral) or informative.

After each block, update the posteriori and read (\mathsf {P} (EV\le 0)).

Threshold (\tau) take 0. 8–0. 9 for a Conservative stop.

D. Risk of ruin/drawdown

Working shares from Kelly (f) (better ⅓ - ½ Kelly).

Set the maximum allowable drawdown to DD (_{\max}) (20-30%).

If the current DD ≥ DD (_{\max}) or conditional RoR ≥ (\beta) (e.g. 5%) - stop.

8) Typical scenarios and ready-made templates

Scenario 1. Positive EV, high volatility (slots, freespins)

(f\approach) ⅓ Kelly; barriers: (T = + 3\s\sigma) session profits, (L = -2\s\sigma).

Every 100-200 spins is a Bayesian check (\mathsf {P} (EV\le 0)).

Any of the three stops works - exit.

Scenario 2. Odds Advantage Bets

Profit/loss barriers in units of rate (e.g. (T = + 10u), (L = -6u)).

SPRT с (\alpha=0. 1,\ \beta=0. 2) between (p_0) (no advantage) and (p_1) (expected).

Drawdown of 20% of the bank is a technical stop.

Scenario 3. New strategy test

Micro bets, limited test pot.

Each (k) events - CI according to (p); if CI includes zero EV → feet, revise hypotheses.

9) Mistakes that break the shutdown

The movement of barriers ("let's move the stop again") - the meaning of probabilistic guarantees is lost.

Ignoring correlations (series, market dependence) - reassessment of the number of independent tests.

Changing the size of the bet without recalculating the rules - the variance/EV changes, the old thresholds are invalid.

Fixing only on profit without confidence and RoR metrics is a high chance of "playing out" to unnecessary drawdown.

10) Bottom line: simple process formula

1. Before start: specify (T), (L), perestroika frequency (k), thresholds (\tau) (for (\mathsf {P} (EV\le 0))), (\alpha ,\beta) (for SPRT), DD ({\max}), (\beta {\text {RoR}}).

2. In-game: After each step/block, check triggers (barriers, EV confidence, RoR/drawdown).

3. If any trigger is triggered, stop without exception.

4. After the session: log - recalculation (p, e ,\sigma), updating thresholds.

If you adhere to these rules, the "moment of stopping by probability" turns from an intuitive pause into a strict management decision: you stop the game exactly when the chance of unfavorable development has become statistically unacceptable - and you retain capital and an advantage for the next, better opportunities.