Probability Basics: Combinations and Why Jackpots Are Hard to Hit
Key Takeaways
- Probability is the ratio of favourable outcomes to total possible outcomes.
- Combinations count selections where order does not matter; permutations count arrangements where order does.
- Lottery draws use combinations because the order numbers are drawn does not affect the result.
- Even moderate increases in pool size cause dramatic increases in the number of combinations.
Probability defined simply
Probability measures how likely an event is. It is expressed as a number between 0 (impossible) and 1 (certain), or equivalently as a fraction or percentage. If you roll a fair six-sided die, the probability of landing on any specific face is 1/6, or about 16.7%.
To find the probability of winning a lottery jackpot, you need two things: the number of winning outcomes (usually 1) and the total number of possible outcomes. The total depends on the game format, and that is where combinations come in.
The core formula is straightforward: Probability = Favourable outcomes ÷ Total possible outcomes. For a jackpot, the numerator is 1 (there is only one winning combination) and the denominator is the total number of unique combinations possible in that game. Everything else in probability analysis for lotteries flows from that simple ratio.
Combinations versus permutations
A permutation is an ordered arrangement. If you care about the sequence in which items appear, you count permutations. A combination is an unordered selection — you only care about which items are chosen, not the order.
Lottery draws are combination problems. Whether the balls emerge in the order 3, 17, 22, 31, 38, 44 or 44, 38, 31, 22, 17, 3 is irrelevant; both represent the same winning set. This distinction matters because combinations produce a smaller count than permutations of the same set, which affects the odds.
To see why the difference is so large, consider picking just 3 items from a pool of 10. The number of permutations is 10 × 9 × 8 = 720. The number of combinations is 720 ÷ 3! = 720 ÷ 6 = 120. Because order does not matter, there are six times fewer combinations than permutations. As the pick count (k) grows, this reduction factor (k!) grows rapidly — for k = 6, it is 720; for k = 7, it is 5,040.
Quick rule of thumb: "Does order matter?" If yes → permutation. If no → combination. Lotteries → combination.
How the combination formula works
The combination formula C(n, k) = n! / (k! × (n − k)!) looks intimidating, but the idea is straightforward. Start with the number of ways to line up k items from a pool of n (that is the permutation count). Then divide by k! to remove the duplicate arrangements that count the same set of items in different orders.
For a concrete example: choosing 6 numbers from a pool of 45 gives C(45, 6). The numerator accounts for the 45 × 44 × 43 × 42 × 41 × 40 possible sequences of 6 numbers. The denominator 6! = 720 removes the re-orderings. The result is 8,145,060 unique combinations.
Worked examples: step by step
Working through a few calculations by hand helps solidify the concept. Below are three examples at increasing scale.
Example 1: Picking 3 from 10. Imagine a small draw where you choose 3 numbers from a pool of 1 to 10. C(10, 3) = 10! / (3! × 7!). The shortcut: 10 × 9 × 8 = 720, divided by 3 × 2 × 1 = 6. Result: 120 combinations. If you hold one ticket, your chance of winning is 1 in 120, or about 0.83%.
Example 2: Picking 5 from 35. C(35, 5) = 35 × 34 × 33 × 32 × 31 ÷ (5 × 4 × 3 × 2 × 1). The numerator is 38,955,840. The denominator is 120. Result: 324,632 combinations. One ticket gives you about a 1 in 324,632 chance — already in the territory where winning is highly unlikely for any individual player.
Example 3: Picking 7 from 45. C(45, 7) = 45 × 44 × 43 × 42 × 41 × 40 × 39 ÷ 5,040. The numerator is 228,713,285,760 and the denominator 5,040 gives 45,379,620 combinations. Compare this to C(45, 6) = 8,145,060. Adding just one more required number to the same pool multiplied the total combinations by a factor of about 5.6.
Worked examples summary
| Game format | Calculation | Total combinations | Odds (1 ticket) |
|---|---|---|---|
| 3 from 10 | C(10, 3) | 120 | 1 in 120 |
| 5 from 35 | C(35, 5) | 324,632 | 1 in 324,632 |
| 6 from 45 | C(45, 6) | 8,145,060 | 1 in 8,145,060 |
| 7 from 45 | C(45, 7) | 45,379,620 | 1 in 45,379,620 |
Why increasing the pool size matters so much
Consider how the total combinations change as the pool grows:
Impact of pool size (picking 6 numbers)
| Pool size | Total combinations | Factor vs 6/40 |
|---|---|---|
| 40 | 3,838,380 | 1× |
| 45 | 8,145,060 | 2.1× |
| 50 | 15,890,700 | 4.1× |
| 55 | 28,989,675 | 7.6× |
| 60 | 50,063,860 | 13.0× |
Adding just 5 more balls to the pool roughly doubles the number of combinations each time. This exponential-style growth explains why high-jackpot games with large pools (such as international draws with 50+ numbers) are extraordinarily difficult to win.
Mathematically, each additional ball adds another multiplicative factor to the numerator while the denominator stays fixed. The net effect compounds quickly.
The role of supplementary and bonus numbers
Some games draw extra numbers (supplementary or bonus balls) to determine secondary prize divisions. These additional draws increase the total number of possible prize outcomes but do not change the jackpot combination count. They allow operators to offer more division tiers — matching 5 main + 1 supplementary, for example — without altering the top-division odds.
In Australian games like Saturday Lotto, two supplementary numbers are drawn after the six main numbers. These do not affect the jackpot, but determine eligibility for Divisions 2 and 5. The supplementary numbers create additional ways to win smaller prizes, while the headline jackpot odds remain determined solely by the main pool and pick count.
Intuition versus calculation
Humans are not naturally good at evaluating very small probabilities. Research in cognitive psychology consistently shows that people overestimate their chances of rare positive events and underestimate common risks. A 1-in-8-million chance feels different from how it calculates. Recognising this gap is not a reason to avoid playing — it is a reason to check the numbers before forming expectations.
Kahneman and Tversky's prospect theory showed that people apply a "probability weighting function" — overweighting very small probabilities and underweighting moderate ones. A 1-in-8-million chance registers in the mind as "unlikely but plausible," when mathematically it is closer to "extraordinarily rare."
This distortion is not a character flaw — it is a feature of how the brain processes uncertainty. Being aware of the gap between intuitive assessment and calculated probability allows you to enjoy participation while keeping financial expectations grounded in reality.
A practical test: before buying a ticket, articulate the odds in a concrete comparison. "My chance of winning this jackpot is about the same as correctly guessing which specific second, out of all the seconds in 94 days, someone is thinking of." That kind of framing activates different reasoning than a raw number like 1 in 8,145,060.
Practical applications beyond lotteries
The combination formula is not exclusive to games of chance. It appears in a wide range of fields, and understanding it through lottery examples gives you a transferable skill.
In medicine, clinical trials often need to test combinations of treatments. If a researcher wants to test 3 drugs chosen from a panel of 12, the number of possible treatment groups is C(12, 3) = 220. Knowing this helps in designing studies and allocating resources.
In information security, password strength depends on the same principles. A system requiring 6 unique characters from 36 alphanumeric options uses permutations, yielding over 1.4 billion arrangements — explaining why longer passwords are disproportionately harder to crack.
In everyday life, choosing a 5-person committee from a 20-person team is a combinations problem: C(20, 5) = 15,504 possible committees. The same mathematics that governs lottery odds applies to organisational decisions, sporting draw structures, and even genetic recombination.
Practical takeaway
Once you understand that combinations define the total outcomes and that pool size drives combinations exponentially, the headline odds of any lottery become transparent. You can verify them yourself with a calculator or a spreadsheet. That transparency supports better decision-making, whether you choose to participate or not.
The combination formula is one of the most universally applicable tools in mathematics. Learning it through the lens of lottery odds equips you to think clearly about probability in many other contexts — from evaluating risk to understanding scientific studies to making informed personal choices.