Problem → Inputs → Model → Outputs → Interpretation

Problem

Detecting roulette bias is a logistical project, not a mystical hunch. A modern advantage player must decide whether a painstaking log of spins contains enough statistical weight to justify wagering into a suspected hot pocket, and if not, how much more evidence is needed. The updated planner elevates this decision from the vague “does zero feel sticky tonight?” to a quantified evaluation of Dirichlet posteriors, Bayes factors, and confidence targets. You can now express the decision in terms of risk thresholds: How certain must the posterior be before you raise stakes? How do different bias magnitudes change the number of observations you need? And what happens to expected value once the casino’s house edge and your bankroll constraints enter the picture?

The new feature set pushes beyond the earlier narrative summary by layering confidence-aware planning, Kelly-style stake sizing, and entropy diagnostics on top of the existing Bayes factor and credible-interval analysis. That means the calculator no longer stops at “is there a bias?” Instead, it helps answer the deeper operational questions: When would the bias become actionable under a target posterior confidence? How do you translate the posterior mean and its tails into bet fractions that protect capital? What is the value of additional logging when the posterior is still nearly uniform? These decision-quality hooks are what separate an actionable advantage plan from an anecdote about streaks.

Because roulette edges are slim, the problem also involves safeguarding against false positives. The planner therefore treats bias detection as a sequential experiment. Every new batch of spins updates the Dirichlet posterior, recomputes the Bayes factor against perfect fairness, and projects how many more observations you would need to clear a user-defined confidence threshold. Those insights create a disciplined stop/go framework: continue logging until the posterior probability that a pocket exceeds a critical margin hits 95%, or until a Bayes factor threshold is met, whichever comes first. Otherwise, stand down and avoid betting into variance.

Inputs

Wheel type toggles between European (37 pockets) and American (38 pockets) layouts. Behind the scenes, that change cascades through every calculation: the uniform baseline probability, the length of the counts vector the parser expects, the payout for the straight-up bet, and the dimension of the Dirichlet prior. Choosing the correct wheel is therefore the very first validation step.

Observed spins per pocket accepts comma- or space-separated integers. The parser enforces that the length of the vector matches the wheel size; a mismatch triggers a descriptive message rather than NaN pollution. These raw counts seed the posterior, drive the marginal likelihood comparison, and populate the CSV export. The helper text reminds you to maintain the canonical order (0, 1, 2, …) so that repeated sessions append cleanly.

Symmetric prior strength (α) lets you encode skepticism. α = 1 implements Laplace smoothing, α = 0.5 approximates Jeffreys’ prior, while higher α values dilute noisy spikes by dragging the posterior back toward uniform. The prior contributes α to every pocket and therefore scales the total pseudo-count α₀ = α × pockets that appears in the posterior variance and the sample-size projections.

Target Bayes factor defines how strong the evidence against a fair wheel must be before you call the bias real. A value such as 30 corresponds to Jeffreys’ “very strong” evidence threshold. The calculator treats this as a planning anchor: it estimates how many more spins you would need, under a hypothesized bias δ, to hit the logarithmic Bayes factor implied by that target.

Target 95% credible interval width controls precision for the most-loaded pocket. If the posterior interval around the leader is still wide, the planner quantifies how many spins remain before the width shrinks below the target. Because the variance of a Dirichlet component scales with the posterior total, this field directly informs how aggressive your sampling schedule must be.

Bias magnitude δ encodes the effect size you care about. The tool assumes one pocket is δ above uniform while the others share the slack. That δ drives two key projections: the Kullback–Leibler divergence used in the Bayes factor forecast and the posterior probability comparisons against baseline and baseline + δ/2.

Target posterior confidence is the newest control. It adds a binary-search solver that finds how many additional spins (if any) are required before the posterior probability that the leading pocket exceeds the critical threshold reaches your desired confidence, e.g., 95% or 99%. This guardrail recognizes that a practical edge hunt cares about posterior risk, not just point estimates.

Model

The backbone remains the conjugate Dirichlet–multinomial update. If the prior is Dirichlet(α) and you observe counts n_i, the posterior parameters become α_i′ = α_i + n_i. Expressed in MathML, the component-wise update is

${\alpha }_i\text{'}={\alpha }_i+n{i}_{}$

Posterior means follow $\frac{{\alpha }_i}{\text{'}}$, and posterior variances use the standard Dirichlet formula that appears in the credible-interval computation. To compare hypotheses, the tool evaluates the marginal likelihood of the counts under the Dirichlet alternative and divides it by the likelihood under the exact-fair multinomial. That Bayes factor, with the combinatorial terms carefully managed for numerical stability, is rendered as

$\mathrm{BF}=\frac{\frac{\Gamma }{(}\frac{\prod }{{\Gamma }_i}}{\frac{\Gamma }{(}\frac{\prod }{{\Gamma }_i}{\left(}^{\frac{1}{p}}}$

To avoid overflow, the implementation relies on a Lanczos log-Γ routine shared with the poker and blackjack tools. Because the log Bayes factor can be extremely large in magnitude, the UI reports the natural log and uses it in downstream planning.

Credible intervals and posterior tail probabilities now use the incomplete beta function via a continued-fraction approximation. The helper rouletteRegularizedBeta implements the algorithm described by Press et al., ensuring that even extreme α and β values (from long spin logs) yield stable probabilities. That routine powers the posterior probability that the leading pocket exceeds the uniform baseline or the stricter baseline + δ/2 hurdle, as well as the new confidence-aware spin planner.

Sample-size projections operate on two fronts. First, the Bayes factor forecast assumes a hypothetical distribution with one pocket biased by δ and others sharing the complement. The required spins are approximated by the ratio of the log target Bayes factor to the KL divergence between that biased distribution and uniform. In MathML, the shortcut is expressed as

$N\approx \frac{\ln B}{\mathrm{KL}(p||u)}$

Second, the target-confidence planner treats the top pocket as a Beta distribution aggregated against “all other pockets”. It incrementally adds spins following a clamped bias assumption until the posterior probability that the pocket clears the threshold exceeds the requested confidence. The bisection uses up to 60 iterations with protective bounds to prevent infinite loops when the target is unattainable. The same solver can therefore reveal that a 0.5% bias might demand tens of thousands of additional spins to satisfy a 99% confidence requirement.

Finally, the calculator translates posterior means into bankroll guidance. A straight-up bet with payout 35:1 has expected value $\mathrm{EV}=35p-(1-p)$. That EV feeds two Kelly fractions: one based on the posterior mean and another using a pessimistic (mean − 1.645 σ) probability. Normalized entropy complements those stake suggestions by summarizing how far the posterior has moved away from a uniform wheel.

Outputs

The results panel opens with Posterior Highlights: total spins logged, the natural log Bayes factor against fairness, and a ranked list of the five hottest pockets with means, 95% intervals, and raw hit counts. Because the list is generated dynamically, it reflects whatever wheel layout and counts you provide.

The Planning Metrics block consolidates the new projections. You see the gap between your current spin total and the Bayes-factor target, the additional spins needed to shrink the top pocket’s credible interval below the requested width, and the bisection solver’s estimate of the extra spins required to hit the posterior confidence threshold. Each statistic automatically clips at zero when your existing data already exceeds the target.

Confidence & Betting translates the posterior into risk-aware statements. Two probability readouts show how likely it is that the top pocket beats the fair-wheel baseline and the stricter baseline + δ/2 bar. The Kelly fractions (mean and pessimistic) quantify recommended bet sizing given your bankroll discipline. Normalized posterior entropy adds context: a value near zero signals a deterministic-looking pocket distribution, while a value near one warns that the wheel still behaves uniformly.

The Expected Value Table compares unit-bet EV under three assumptions: a fair wheel, a half-δ bias, and a full δ bias. This table, combined with the probability outputs, highlights how house edge turns positive only once a meaningful bias materializes. The CSV export mirrors these insights with columns for posterior means, 95% intervals, the probability of beating fairness, and the probability of beating the δ/2 threshold, enabling deeper offline analysis.

Interpretation

The composite picture encourages disciplined action. A strongly negative log Bayes factor, high entropy, and low probabilities above the δ threshold all argue for continued observation rather than betting. Conversely, a posterior that simultaneously shows a positive EV, manageable entropy, and a shortfall of only a few hundred spins to reach a 95% confidence suggests the wheel may be ripe for exploitation.

Remember that the Bayes-factor projection assumes the bias you enter actually exists. If you suspect a smaller δ, the same UI instantly illustrates how dramatically the required spins expand. Likewise, the Kelly fractions should be interpreted as ceilings: even when the posterior mean suggests a 0.6% stake, the pessimistic fraction might halve that recommendation, reflecting the risk of overfitting to noise.

The hash serialization means every input state can be shared with a teammate by copying the URL. Accessibility affordances—labels, helper text, and the aria-live results region—ensure the planner remains usable during long sessions, even on mobile devices.

Worked Example: 10,059 European Spins with a Sticky Zero

Suppose you log 10,059 spins on a European wheel and observe the counts preloaded in the form. Pocket 0 appears 340 times, while most numbered pockets land around 268–274 hits. With α = 1, δ = 0.02, target Bayes factor 30, target credible width 0.005, and target posterior confidence 95%, the calculator delivers the following diagnostics:

• Total spins: 10,059. The natural log Bayes factor against fairness is approximately −36,277, indicating that, despite the noisy excess on zero, the uniform model still dominates given the symmetric prior.
• Pocket 0’s posterior mean is 0.0338 with a 95% interval of roughly [0.030, 0.037]. Five other pockets cluster near the baseline 0.0270, reinforcing how narrow the edge might be.
• The probability that pocket 0 exceeds the fair-wheel baseline exceeds 99.99%, yet the probability it beats baseline + δ/2 (≈ 0.0370) is only about 3.8%. That contrast warns that “better than fair” is not the same as “actionably biased.”
• The planning metrics report that you have already logged more spins than the 544 implied by the δ = 0.02 Bayes-factor target, but you still need about 9,963 additional spins to drive the 95% interval width below 0.005 and roughly 7,471 more spins to hit 95% confidence that the pocket clears the δ/2 hurdle.
• The posterior mean Kelly fraction is approximately 0.62% of bankroll per spin, while the pessimistic fraction (using mean minus 1.645σ) drops to about 0.31%. Normalized entropy clocks in at 0.9998, screaming that the distribution remains nearly uniform despite the apparent hot pocket.
• In the expected-value table, a unit bet on pocket 0 has EV ≈ +0.216 units if the posterior mean were exact, compared with −0.027 units on a fair wheel and +0.693 units if the full δ bias were real.

Viewed together, the metrics argue for patience. The Bayes factor and entropy agree that your evidence is insufficient; the confidence planner quantifies the remaining grind; and the conservative Kelly fraction caps the bankroll exposure even if you choose to dabble in small probes.

Comparison Table: Evidence Scenarios

The table illustrates why tiny deviations demand huge samples. Halving δ from 0.02 to 0.01 nearly quadruples the Bayes-factor requirement and the confidence-driven spin budget. By δ = 0.005, you are effectively committing to a multi-session engineering project before probability thresholds catch up.

Assumptions, Limitations, and Practical Tips

The model assumes independent spins and a stationary bias. Temperature shifts, rotor swaps, or ball wear can invalidate that assumption by changing probabilities mid-log. The normal approximation for credible intervals works best when counts exceed about 30 per pocket; otherwise, rely more heavily on the posterior probabilities that use the exact Beta function.

Because the target-confidence solver clamps probabilities and iteration counts, extremely ambitious configurations (δ near zero, confidence 0.9999) may return “N/A” when the bias is effectively undetectable. Treat that as a signal to rethink the edge size rather than a failure of the tool.

Practical tips: (1) Normalize your data entry process—record spins in pocket order immediately to avoid transcription errors. (2) Use the CSV export to monitor trend lines session by session; the new probability columns make it easy to plot how confidence evolves. (3) Stress-test multiple δ values before a casino trip to understand how sensitive your plan is to overestimating the bias. (4) Pair the Kelly fraction with an absolute bankroll stop so that even surprise cold streaks cannot erase prior profits. (5) When the normalized entropy refuses to budge, switch wheels; the data is telling you the bias is either nonexistent or currently masked.

Testing Checklist

• Edge cases: mismatched count lengths, zero prior, or δ ≤ 0 trigger human-readable validation messages without breaking the result panel.
• Rounding: posterior means print to four decimals, intervals clamp to [0, 1], and probability displays format as percentages with two decimals.
• Performance: verified responsiveness with 100,000 spins and confidence targets up to 0.999; the continued-fraction beta function stayed stable.
• Accessibility: form controls remain labeled, the aria-live region announces updates, and keyboard navigation covers hash-serialization and CSV download flows.