Feasible support
The smallest possible overlap is max(0, K − (N − n)); the largest is min(K, n). Impossible match counts are excluded rather than displayed as misleading zeros.
Probability
Lottery Distribution Calculator
Calculate the exact hypergeometric probability of every lottery match count, cumulative target odds, expected hits, and expected ticket spend without simulation.
EXACT LOTTERY DISTRIBUTION
Account for every possible match count before interpreting the odds
This page enumerates the complete match distribution for a choose-without-replacement lottery. It separates the probability on one ticket from the expected count across many ticket evaluations, so “one in” odds are not confused with a guarantee.
Exact target probability—
Target or better—
At least one target in t tickets—
Expected exact-target hits—
Tickets per target in expectation—
Expected spend per target—
EXACT LOTTERY DISTRIBUTION
Complete exact match distribution
Use the complete ledger to distinguish an exact tier from “that tier or better,” then convert a single-ticket probability into an expectation or an at-least-one probability only for a declared ticket count.
| Matches | Favorable tickets | Probability (%) | One in | Expected count | P(matches or more) (%) |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
P(X = r) = C(K, r) × C(N − K, n − r) ÷ C(N, n)
The hypergeometric model counts tickets with r winning selections and n − r non-winning selections, then divides by every possible n-number ticket. It is exact when the pool, draw, and ticket all contain distinct numbers and every valid draw is equally likely.
Waiting for valid inputs.
HOW TO USE THIS MODEL
Describe the lottery rules before asking what a match tier means
- Enter the total pool size; bonus balls or separate pools require a different model.
- Enter how many distinct main numbers the draw selects.
- Enter how many distinct main numbers appear on one ticket.
- Choose the exact match tier you want to inspect.
- Enter a ticket count only if you need an expected frequency or at-least-one probability.
- Review the full ledger and confirm that all match probabilities reconcile to 100%.
EXACT LOTTERY DISTRIBUTION FUNDAMENTALS
Six distinctions that keep lottery odds honest
- Without replacement
- A number cannot be selected twice in the same ticket or main draw; this dependence is why the binomial model is not exact.
- Exact versus cumulative
- Exactly four matches excludes five and six; four or more includes all three tiers.
- Probability versus expectation
- An expected 0.2 hits is a long-run average, not a claim that part of a win occurs.
- One-in notation
- One in 10,000 is the reciprocal of a one-ticket probability, not a schedule for when a result must happen.
- Multiple tickets
- The at-least-one formula assumes independent ticket evaluations with the same target probability. Overlapping selections can change dependence.
- Prize model omitted
- Match probability alone does not establish payout, taxes, shared jackpots, expected return, or affordability.
MODEL AND FORMULA
Read the equation before interpreting the result
P(X = r) = C(K, r) × C(N − K, n − r) ÷ C(N, n)
The hypergeometric model counts tickets with r winning selections and n − r non-winning selections, then divides by every possible n-number ticket. It is exact when the pool, draw, and ticket all contain distinct numbers and every valid draw is equally likely.
DEEP PROBABILITY ANALYSIS
Why the exact distribution is more useful than a jackpot headline
Combinatorial denominator
C(N,n) is the set of all valid distinct tickets. Each numerator selects r winners and fills the remaining ticket positions from non-winners.
Repeated-ticket interpretation
Expected hits scale linearly as t×p, while the chance of at least one is 1−(1−p)^t. Those answer different planning questions.
WORKED DECISION CASES
Two decisions the exact table can support
Checking a published prize tier
Enter the operator’s main-number rules and compare the exact-tier reciprocal with the published odds. Treat bonus-number mechanics as a separate event, not as another main-number match.
Teaching rare-event scale
Hold the game rules fixed and increase the ticket count. Expected hits rise linearly, but no finite purchase count turns a random target into a certainty.
PROBABILITY GLOSSARY
Terms used in the distribution ledger
- Population N
- Total distinct values from which the draw is made.
- Winning set K
- Distinct main numbers selected by the draw.
- Ticket sample n
- Distinct values selected on one ticket.
- Match count r
- Size of the overlap between ticket and winning set.
- Combination C(a,b)
- Number of b-element subsets that can be chosen from a elements.
- Probability mass
- Probability assigned to one discrete match count.
- Cumulative tail
- Sum of probabilities at or above a chosen match count.
- Expected frequency
- Ticket count multiplied by the one-ticket probability.
EVIDENCE AND DATA LINEAGE
What to retain when checking published lottery odds
Save the game rules, draw date, pool size, number of main balls, ticket selection size, treatment of bonus balls, target-tier definition, ticket count, ticket price, and the unrounded probability table. If rules change, preserve the version effective for the draw being analyzed.
LIMITS AND EXCLUSIONS
What this analytic model does not establish
- It does not handle bonus-ball, Powerball-style second-pool, ordered-number, replacement, or duplicate-number rules.
- It assumes each valid main-number draw is equally likely and the stated rules are implemented correctly.
- It does not calculate prize value, jackpot sharing, taxes, expected monetary return, or gambling affordability.
- The repeated-ticket at-least-one result assumes independent evaluations with unchanged odds.
- Expected tickets per target is a reciprocal probability, not a prediction of when a hit will occur.
RELIABLE SOURCES
References for the probability model and its limits
FREQUENTLY ASKED QUESTIONS
Questions about exact lottery match probabilities
Why is this hypergeometric rather than binomial?
Lottery numbers are drawn without replacement, so match opportunities within one draw are dependent. The hypergeometric model counts that structure exactly.
Does “one in 1,000” mean I will win within 1,000 tickets?
No. It is a reciprocal probability. Random sequences can contain long gaps or clusters.
Why does exactly four differ from four or more?
The cumulative event also includes five and six matches; the exact event does not.
Can I add a bonus ball to K?
Not safely when the bonus ball has different prize rules. Model the joint event explicitly.
Do multiple lines on one draw count as independent?
Not necessarily. Shared or deliberately covered numbers create dependence even though each line has the same marginal odds.
Why can expected hits be below one?
Expectation averages the count across many repetitions. A single batch still produces whole-number outcomes.
Does a larger spend improve expected value?
It increases exposure to the same odds. Expected monetary value also requires prize amounts, prize sharing, and costs, which this page deliberately omits.
IMPORTANT PROBABILITY NOTE
Probability is not a spending recommendation
This educational model explains main-number match probabilities. It does not recommend gambling, predict a draw, guarantee a result, or replace the operator’s official rules and published prize table. Set spending limits independently of calculated odds.
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