Compute exact lottery odds for standard number-drawing games. Define the game rules and this
calculator will return the probability of matching 0, 1, 2, … up to all numbers, plus the
familiar “1 in X” jackpot odds.
The engine uses hypergeometric probability behind the scenes and can handle both
single-drum lotteries (classic 6-from-49 style) and two-drum lotteries with bonus balls
(Powerball-style games).
1. Choose lottery type
Single-drum: all numbers are drawn from the same pool.
Two-drum: main numbers + separate bonus/Powerball pool.
2A. Single-drum lottery parameters
Example: “Choose 6 numbers out of 49; draw 6 winning numbers out of the same 49”.
Total distinct numbers in the game (e.g. 49).
How many distinct numbers you choose.
How many numbers the lottery draws.
Used for “P(matches ≥ rmin)” in the summary.
2B. Two-drum lottery parameters
Example: “Pick 5 main numbers from 1–69 and 1 bonus ball from 1–26; draw 5 main + 1 bonus”.
Main number pool
Bonus / Powerball pool
Exact combinatorial odds – no approximations
Single-drum lottery odds
Summary
Pool size (N)
Ticket numbers (T)
Numbers drawn (K)
Total combinations C(N, K)
Jackpot matches
Jackpot probability
Jackpot odds (“1 in X”)
P(matches ≥ rmin)
Interpretation
All calculations assume a perfectly random draw without replacement.
Odds refer to one ticket in a single drawing.
Probability of exactly r matches
Matches r
P(exactly r)
“1 in X” odds
For a given r, the probability is
\( \mathbb{P}(R = r) = \dfrac{\binom{T}{r}\,\binom{N - T}{K - r}}{\binom{N}{K}} \),
where R is the number of matches between your ticket and the drawn K numbers.
Two-drum lottery odds
Summary
Main pool (N₁)
Main ticket (T₁)
Main drawn (K₁)
Bonus pool (N₂)
Bonus ticket (T₂)
Bonus drawn (K₂)
Jackpot probability
Jackpot odds (“1 in X”)
Interpretation
The calculator assumes that the main and bonus pools are drawn independently.
Overall odds are the product of the odds in each drum.
Joint probability table – main matches vs bonus matches
Each cell corresponds to
\( \mathbb{P}(R_1 = r_1, R_2 = r_2) = \mathbb{P}(R_1 = r_1)\,\mathbb{P}(R_2 = r_2) \),
where \( R_1 \) and \( R_2 \) are the numbers of matches in the main and bonus pools respectively.
The jackpot is the cell with all main and all bonus numbers matched.
Lottery odds and hypergeometric probability
Most number-drawing lotteries can be modelled using the
hypergeometric distribution. Consider a single-drum game:
there are N possible numbers, you choose T of them on your ticket, and K numbers are drawn
for the winning combination. The number of matches between your ticket and the draw, R, follows
a hypergeometric law.
where \( \binom{n}{k} \) is the binomial coefficient “n choose k”.
The denominator \( \binom{N}{K} \) counts all possible K-number draws from N,
while the numerator counts favourable draws with exactly r matches.
Jackpot odds as “1 in X”
For the jackpot, r is typically equal to all numbers on your ticket (e.g. r = 6 in a 6-from-49
game). Once the jackpot probability \( p_{\text{jackpot}} \) is known, the usual way to present odds is
\[
\text{Odds} = \frac{1}{p_{\text{jackpot}}}.
\]
If \( p_{\text{jackpot}} \approx 7.15 \times 10^{-8} \), the odds are approximately
\( 1 / (7.15 \times 10^{-8}) \approx 13{,}983{,}816 \), so we say the jackpot odds are about
“1 in 13,983,816”.
Two-drum lotteries with bonus balls
In a two-drum game, the main numbers and the bonus balls are drawn from separate pools.
If \( R_1 \) is the number of matches in the main pool and \( R_2 \) in the bonus pool, then:
because the two drums are independent. The jackpot corresponds to
matching all main numbers and all bonus numbers at the same time.
Practical notes and limitations
The calculator works with exact integer combinatorics implemented in floating-point
arithmetic. For typical lottery sizes (N up to ~80) this is numerically stable.
Real lotteries often define multiple prize tiers (e.g. “3+1”, “4+0”, “5+1”). You can derive
the odds for each tier from the joint probability table of matches.
Buying more tickets scales your probability linearly but does not change how small the
base probability is – a key point in responsible gambling education.
Lottery odds – FAQ
Lotteries are designed so that the number of possible winning combinations is enormous.
For example, in a 6-from-49 game there are C(49, 6) = 13,983,816 possible draws.
Your ticket corresponds to exactly one of these combinations, so the jackpot odds for
a single ticket are 1 in 13,983,816.
In a properly designed lottery, quick-pick and self-chosen numbers have the
same mathematical odds of winning. Quick picks can reduce patterns and
shared numbers across many players, but they do not change your single-ticket chance
of hitting the winning combination.
For standard lotteries, the expected monetary value of a ticket is usually far below
its price once you factor in all prize tiers and the fact that the organizer takes
a significant share. Lottery tickets are best seen as entertainment, not as an
investment strategy.
Yes. The tool is ideal for teaching combinations, hypergeometric probability,
and risk literacy. Students can adjust N, T and K to model different games and
immediately see how small changes in the rules drastically affect jackpot odds.