this page How likely each tier is. A count of combinations divided into the total, fixed by the format and identical for every player. Nothing about the ticket, the operator or the week changes it.
Draw Desk / The odds
Draw 03
Every tier’s odds come from one figure
The chance of a tier is not a published number to be taken on trust. It is a count of combinations, and one figure decides it: the number of ways to choose six numbers out of fifty-nine. This page derives every tier from that figure, so each number can be re-derived rather than believed.
Draw ticket
- Sheet
- DD-57-04
- Subject
- The odds
- Top-tier odds
- 1 in 45,057,474
- Any prize
- 1 in 92
3 tiersA printed sum: the rules state the amount, and it pays whoever matched, however many that is
1 tierA split fund: a declared share of the prize fund, divided among the lines that matched it
1 tierCarried forward: an unwon share of the fund moved into a later draw instead of paid
Direct answerIn a draw where six numbers are selected from fifty-nine, there are 45,057,474 possible combinations, so a single line matches the top tier with probability one in 45,057,474. Each lower tier is a count of the other combinations: 318 lines match five, 20,670 match four, 468,520 match three, and 489,509 lines match three or more, which is a chance of about one in ninety-two of winning anything.
One figure decides itC(59,6) = 45,057,474Tiers are countsAny prize: 1 in 92
The combination that sets the top tier
The chance of matching all six numbers is the number of ways to choose six of fifty-nine considered as a set, drawn without replacement and without regard to order. That is the binomial coefficient, and it is the only figure needed.
Every other tier is a variant on the same count. A line matching exactly five of the six drawn numbers has one number among the fifty-three that were not drawn, so there are fifty-three such lines for each choice of which five of the six were matched, and six ways to choose which five. The product is the count of that tier, and the same construction gives every tier below it.
Every tier, derived
- Top tier - match 6. One way to match all six: C(6,6) x C(53,0) = 1, so the chance is 1 in 45,057,474.
- Match 5. Choose which five of the six are matched (6 ways) and one of the remaining fifty-three (53 ways): 6 x 53 = 318, so the chance is 45,057,474 / 318 = 1 in 141,690.
- Match 4. Choose four of six (15 ways) and two of the remaining fifty-three (1,378 ways): 15 x 1,378 = 20,670, so the chance is 45,057,474 / 20,670 = 1 in 2,180.
- Match 3. Choose three of six (20 ways) and three of the remaining fifty-three (23,426 ways): 20 x 23,426 = 468,520, so the chance is 45,057,474 / 468,520 = 1 in 96.
- Match 2. Choose two of six (15 ways) and four of the remaining fifty-three (292,825 ways): 15 x 292,825 = 4,392,375, so the chance is 45,057,474 / 4,392,375 = 1 in 10.3.
Worked example - the top tier, and the chance of any prize (illustrative)
Draw 6 of 59. Total combinations: C(59,6)
C(59,6) = 59 x 58 x 57 x 56 x 55 x 54 / (6 x 5 x 4 x 3 x 2 x 1)
= 32,441,381,280 / 720
= 45,057,474
Tickets matching each tier and the chance of one line:
match 6: 1 of 45,057,474 = 1 in 45,057,474
match 5: 318 of 45,057,474 = 1 in 141,690
match 4: 20,670 of 45,057,474 = 1 in 2,180
match 3: 468,520 of 45,057,474 = 1 in 96
match 2: 4,392,375 of 45,057,474 = 1 in 10.3
Any prize (match 3 or better):
1 + 318 + 20,670 + 468,520 = 489,509 combinations
45,057,474 / 489,509 = 1 in 92
The last line is the one readers most often get wrong. A prize of some kind is
roughly a one-in-ninety-two event per line, which is why the smallest tiers are
frequent and why the product feels generous - and it is set entirely by the counts above.
What the odds do not say
Two riders belong with any odds figure, and both are arithmetic rather than opinion.
- Odds are per line, and lines add. Ten lines have ten times the chance of one, at ten times the price. There is no discount for a larger purchase and no smart way to arrange a set of lines: every arrangement of the same number of distinct lines covers the same count of combinations.
- The odds are independent of the amount. How much a tier pays is a separate question from how likely it is. The tiers page is about the amount, and this page is about only the chance - the two are set by different things and a large prize does not imply a likely one.
Odds and return are two different figures
the odds
the return
the fund What the tiers pay, on average. Set by the declared split, not by the tier probabilities. The prize fund page shows a sample draw returning half of what was staked, which is a different number from any of the odds above.
the rollover
the exception One draw can carry extra money. When a top tier is unwon its share moves to a later draw, which raises the effective return of that particular draw. It does not change the odds, the split, or the return of an ordinary draw. The rollover page works this out in full.