Zero wins are still informative
With x = 0, Wilson produces a zero lower bound and a positive upper bound. The data constrain the rate but cannot prove the true probability is zero.
Probability
Lottery Confidence Calculator
Calculate a Wilson confidence interval for an observed lottery win rate, compare wins with declared probability, estimate at-least-one-win probability, and separate sampling uncertainty from expected payoff.
REPEATED-DRAW UNCERTAINTY
Put a defensible interval around the observed win rate
This calculator treats each recorded ticket as a Bernoulli trial, builds a Wilson score interval around the observed winning proportion, and compares the count with a declared per-ticket probability. It is designed for audit and simulation review—not for claiming that recent wins change future draw odds.
Observed win rate-
Wilson lower bound-
Wilson upper bound-
Expected wins at declared rate-
Count difference (SD)-
Chance of at least one win-
REPEATED-DRAW UNCERTAINTY
Win-rate uncertainty and declared-odds ledger
Use the interval to describe the precision of an observed or simulated win rate. Compare it with the declared probability only when every trial shares the same game, prize definition, and sampling process. Apparent disagreement can reflect sampling noise, data selection, dependence, or a mismatched win definition.
| Diagnostic | Observed or declared input | Scale term | Calculated value | Unit / meaning |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
p̂ = x/n; Wilson = [p̂ + z²/(2n) ± z√(p̂(1−p̂)/n + z²/(4n²))] / [1 + z²/n]
x is the recorded win count, n is the number of comparable tickets, z is the two-sided normal critical value for the selected confidence level, and p̂ is the observed proportion. A separate binomial expectation np0 uses the declared probability p0; the Wilson interval does not replace an exact game-odds calculation.
Intermediate values remain unrounded until display formatting.
HOW TO USE THIS MODEL
Define the win event before calculating an interval
- Choose one event—such as any paid prize or one specific prize tier—and count only tickets assessed by that definition.
- Enter the number of comparable tickets and winning tickets; keep different games, rule versions, and promotional entries separate.
- Enter the official probability for the same event as a percentage, not an odds denominator.
- Select the confidence level; higher confidence creates a wider interval because it demands greater coverage.
- Interpret the Wilson bounds together with expected wins, count standard deviation, and data-lineage limits.
REPEATED-DRAW UNCERTAINTY FUNDAMENTALS
What a win-rate confidence interval can and cannot say
- Observed proportion
- The sample rate x/n summarizes recorded outcomes. It is not automatically the official probability or a forecast for the next ticket.
- Wilson score interval
- Wilson bounds remain meaningful for small counts and zero wins, where a simple plus-or-minus normal interval can collapse or leave the valid 0–1 range.
- Confidence level
- A 95% method describes long-run interval coverage under repeated comparable samples; it does not assign a 95% probability to one already-calculated fixed interval.
- Declared probability
- The comparison value must refer to the same game and win definition. “Any prize” cannot be compared with jackpot-only odds.
- Independence
- The binomial model assumes trials have the same probability and do not change one another. Pools, correlated records, and duplicated tickets can violate that assumption.
MODEL AND FORMULA
Wilson interval plus a separate binomial count diagnostic
p̂ = x/n; Wilson = [p̂ + z²/(2n) ± z√(p̂(1−p̂)/n + z²/(4n²))] / [1 + z²/n]
x is the recorded win count, n is the number of comparable tickets, z is the two-sided normal critical value for the selected confidence level, and p̂ is the observed proportion. A separate binomial expectation np0 uses the declared probability p0; the Wilson interval does not replace an exact game-odds calculation.
DEEP PROBABILITY ANALYSIS
Why sample design matters more than extra decimals
Rare events converge slowly
When the declared probability is tiny, thousands of trials may still yield no events. Large relative uncertainty is a property of the evidence, not a formatting problem.
Selection can dominate sampling error
Recording only sessions with wins, mixing free plays with paid tickets, or stopping after a success changes the sample. A narrow interval cannot repair biased collection.
WORKED DECISION CASES
Two legitimate uses of the confidence model
Validate a lottery simulation
Run a fixed number of simulated tickets, count outcomes under a documented prize definition, and check whether the declared rate is plausible relative to the Wilson interval and standardized count difference.
Audit a promotional claim
A reviewer can compare the observed any-prize rate in a complete campaign log with the pre-published rate, provided ticket eligibility, redemptions, and excluded records are retained.
PROBABILITY GLOSSARY
Sampling terms used on this page
- Bernoulli trial
- One trial recorded as event or non-event under a fixed definition.
- Observed rate
- Winning tickets divided by tickets observed.
- Wilson interval
- Score-based interval estimate for a binomial proportion.
- Critical value
- Normal quantile z corresponding to the selected two-sided confidence level.
- Expected count
- n multiplied by the declared per-ticket probability.
- Standardized difference
- Observed minus expected count divided by the binomial count standard deviation.
EVIDENCE AND DATA LINEAGE
Keep the trial definition and complete record sequence
Retain game and rule version, drawing or simulation seed, ticket identifiers, event definition, eligible and excluded records, declared odds source, ticket count, win count, confidence level, prize basis, cost basis, collection window, and any stopping rule. Without that lineage, a precise-looking interval can describe the wrong sample.
LIMITS AND EXCLUSIONS
Conditions that block a confidence claim
- The Wilson interval assumes comparable Bernoulli trials; it does not validate a random-number generator or detect every form of dependence.
- The declared-odds comparison is invalid when the observed win definition differs from the official probability definition.
- The standardized count difference is a screening diagnostic, not a complete hypothesis test or evidence of fraud.
- Expected net payoff uses one average prize and omits prize-tier variance, taxes, cash-option treatment, and jackpot sharing.
RELIABLE SOURCES
References for this page's method and boundaries
FREQUENTLY ASKED QUESTIONS
Lottery confidence and sampling questions
Why use Wilson instead of p̂ plus or minus z standard errors?
The simple Wald interval performs poorly with small samples or proportions near zero or one. Wilson incorporates the score-test geometry and stays within the valid probability range after bounding.
Can I enter jackpot wins and any-prize probability?
No. The numerator and declared probability must describe the same event. Comparing jackpot wins with any-prize odds produces a mathematically calculated but meaningless diagnostic.
What does it mean if the official probability is outside the interval?
It flags tension between the observed sample and declared model under the stated assumptions. Check event definitions, exclusions, dependence and data collection before treating the difference as substantive.
Why is the upper bound positive when there were zero wins?
Zero observed events do not prove zero probability. The upper bound represents rates still compatible with a finite zero-event sample at the selected confidence level.
Does a recent losing streak increase the next ticket’s chance?
No under an independent fixed-odds model. The confidence interval summarizes the recorded sample; it does not create memory in the next draw.
Is the expected-net line a profitability test?
No. It uses the declared win probability and one average prize, so it omits the prize distribution and other lottery-specific adjustments needed for a full expected-value model.
IMPORTANT PROBABILITY NOTE
Do not turn sample uncertainty into a gambling prediction
This calculator is educational statistical software, not a detector of rigging and not gambling, legal, financial, or tax advice. Use complete comparable records and current official odds. A confidence interval cannot justify chasing losses or exceeding an entertainment budget.