1. The probability model TZOKER uses
TZOKER follows a hypergeometric distribution. Unlike rolling a die (where each event is independent), draws pick numbers without replacement: a number that has already been drawn cannot come up again in the same draw.
2. The formula
The probability of matching exactly k of the 5 main numbers is:
P(k of 5) = [C(5, k) × C(40, 5−k)] / C(45, 5)
where C(n, r) is the binomial coefficient (n choose r). The denominator C(45, 5) = 1,221,759 is the total number of possible 5-number combinations from 1–45.
For the Tzoker bonus number (1–20), probability is independent: 1/20 = 5% to hit it, 19/20 = 95% to miss it.
3. Combining with the Tzoker
Because the two fields are independent, the combined probability for each tier is the product:
P(k mains + Tzoker) = P(k of 5) × (1/20)
P(k mains, no Tzoker) = P(k of 5) × (19/20)
4. Worked example: 5+1 (jackpot)
P(5 of 5) = C(5,5) × C(40,0) / C(45,5) = 1 / 1,221,759.
P(5+1) = (1/1,221,759) × (1/20) = 1 / 24,435,180.
5. Full odds table
| Tier | Match | Probability | 1 in … |
|---|---|---|---|
| 1 | 5 + 1 | 0.0000041% | 24,435,180 |
| 2 | 5 | 0.0000778% | 1,286,062 |
| 3 | 4 + 1 | 0.0008185% | 122,176 |
| 4 | 4 | 0.0155513% | 6,430 |
| 5 | 3 + 1 | 0.0319212% | 3,133 |
| 6 | 3 | 0.6065026% | 165 |
| 7 | 2 + 1 | 0.4043351% | 247 |
| 8 | 1 + 1 | 1.8700497% | 53 |
| 9 | 2 | 7.6823662% | 13 |
Overall odds of winning any prize: 1 in ~9.4 (~10.6%).
Only tier 1 (5+1) is distributed pari-mutuel, and when nobody matches it the amount goes to a rollover into the next draw. Tiers 2 to 9 pay fixed, predefined amounts that do not depend on ticket sales.
6. Systems (picking more than 5 numbers)
Picking 6 numbers instead of 5 plays C(6, 5) = 6 columns. Picking 7 plays C(7,5) = 21 columns, and so on. Cost grows linearly, but the probability of hitting at least one 5-match grows proportionally. The number generator computes exact cost and odds automatically for any selection size.
7. Common mistakes
- "Number X is overdue": Wrong. Each draw is independent — past results don't influence the next event (the gambler's fallacy).
- "Hot numbers have better odds": Wrong. A 1–2 year sample is not statistically meaningful for a sample space this large; observed frequency converges to 1/45 in the long run.