How lottery odds work

1. Counting combinations

In Saturday Lotto you choose 6 numbers from 45. Order doesn't matter, so 3-17-22-30-41-44 is the same game as 44-41-30-22-17-3. The number of different games is “45 choose 6”:

C(45,6) = 45! ÷ (6! × 39!) = (45×44×43×42×41×40) ÷ (6×5×4×3×2×1) = 8,145,060

Exactly one of those games matches the six winning numbers, so Division 1 is 1 in 8,145,060. Oz Lotto's C(47,7) = 62,891,499 and Set for Life's C(44,7) = 38,320,568 work the same way.

2. Matching some of the numbers

To match exactly 4 of 6 winning numbers in Saturday Lotto, your game needs 4 numbers from the 6 winners and 2 from the 39 losers. Multiply the ways: C(6,4) × C(39,2) = 15 × 741 = 11,115. Divide by 8,145,060 and you get Division 4's odds of about 1 in 733.

Mathematicians call this the hypergeometric distribution: drawing without replacement from a population that has two kinds of item.

3. Supplementary numbers

Supps split the losing balls into two groups: the supps and everything else. A division like “5 winning numbers + a supp” needs 5 winners and 1 supp: C(6,5) × C(2,1) = 12 games, or 1 in 678,755. When a division needs “at least one” supp across several slots, it's easiest to count all the ways to fill those slots and subtract the ways that use no supp. Each game page shows that working.

4. Powerball's second barrel

With two independent barrels you multiply. Australian Powerball has C(35,7) = 6,724,520 main-number combinations and 20 Powerballs, giving 134,490,400 outcomes. Any main-number result happens with the right Powerball 1 time in 20 and the wrong one 19 times in 20.

5. System entries

A System 8 in Saturday Lotto covers every 6-number game inside your 8 numbers: C(8,6) = 28 games. Its Division 1 odds are 28 times better, at 28 times the price. Systems are a convenient way to buy overlapping games. They don't beat the odds. On average you win the same number of games as you would from 28 separate entries, but the wins come in clusters.

That clustering is why syndicates like systems: one big system means a single ticket to check and shared, clustered wins. If you want to spread your coverage instead, look up covering designs, the branch of combinatorics that asks for the fewest games that guarantee a certain match somewhere in your set.

6. “Any prize” odds

Headline figures like “1 in 42 to win a prize” are real, but almost all of those wins are the lowest division, which often pays less than a few games cost. The chance of winning a prize worth more than your ticket is much lower than the headline.

7. Expected value

Lotteries return a fixed share of sales as prizes and keep the rest for operating costs, taxes and duties. So the average return on every dollar spent is well under a dollar. No number-picking strategy changes that, because every combination has the same chance.

8. The gambler's fallacy

“Number 17 is overdue” and “those numbers just won, they won't come up again” are both wrong. Each draw is independent. A number that hasn't appeared for months has exactly the same chance as one drawn last week. The one useful idea here is avoiding popular combinations: it doesn't help you win, but it means sharing less if you do.

Try it yourself: Saturday Lotto · Oz Lotto · Powerball · Set for Life