How difficult is a Lotto jackpot to win? – part 1
Undoubtedly, winning a Lotto jackpot is challenging. While accepting it may not be easy, how difficult can it be?
In one respect, it is straightforward. You must buy a ticket comprising at least one board in the next Lotto draw to stand any chance.
Such a trite response has little benefit except to present an opportunity. You can only win if you take part. However, we shall make progress only when we examine the underlying mathematics to gain valuable insights.
To complicate matters, Camelot changed Lotto in October 2015 after 2,065 draws, making it harder to win a jackpot. Afterwards, the game would use 59 balls instead of 49 and have a different prize structure.
In the first part of this article, we shall consider only the earlier version of Lotto, played with 49 balls.
Table of contents
Estimated reading time: 8 minutes
Original game with 49 balls
The following discussion relates to the original game played with 49 balls over the first 2,065 draws.
The results are independent of the order in which the lottery balls are drawn, meaning we must consider combinations instead of permutations.
Combinations
Mathematically speaking, there are 49C6 combinations when choosing any six from 49 numbers, like Lotto balls in the original game, for instance.
$$ \begin{split}^{49}C_6& = \frac{49!}{6! × 43!}\\ \\&=\frac{(49×48×47×46×45×44×43×\dotso×6×5×4×3×2×1)}{(6×5×4×3×2×1)×(43×42×41×\dotso×3×2×1)}\\ \\&=13983816 \end{split} $$The exclamation mark ! indicates a factorial number, and the ellipsis … means and so on until. e.g. 6! ≡ 6×5×4×3×2×1 = 720.
Many factors in the numerator and denominator above cancel out.
$$ ^{49}C_6 ≡ \binom{49}{6} $$As you can see, it is tedious to write out the process in longhand; the abbreviation above makes things clearer.
Applying the maths
Since each combination consists of 6 of the 49 balls, it follows each ball appears (13,983,816 × 6) ÷ 49 times. i.e. 1,712,304 times. This fact may be interesting, but it is not particularly helpful in our quest for success; we need better insights.
The total number of combinations is slightly less than 14 million. Only one contains all six main numbers. What about the rest? How many of the others are prize-winning combinations?
The following table shows how many combinations match the main numbers drawn.
| Category | Combinations | Combinations (%)[1] | Winning Combinations (%) |
|---|---|---|---|
| No Match | 13,723,192 | 98.14 | 0 |
| Match 3 | 246,820 | 1.77 | 94.70 |
| Match 4 | 13,545 | 9.69×10-2 | 5.20 |
| Match 5 | 252 | 1.80×10-3 | 9.67×10-2 |
| Match 5+ | 6 | 4.29×10-5 | 2.30×10-3 |
| Match 6 | 1 | 7.15×10-6 | 3.84×10-4 |
| Grand Total: | 13,983,816 | 100% | 100% |
In the case of Match 5+, it is straightforward to see that six combinations are possible since each of the six main numbers can be replaced by the bonus number.
To summarise, only 260,624 (1.86%) of the 13,983,816 combinations possible win prizes. In other words, over 98% of them win nothing at all! The pie chart on the left below demonstrates this vividly; winning combinations account for the minuscule blue slice.
Most of the winning combinations (almost 95%) qualify only for the minimum Match 3 prize; about 5% match four of the six main numbers. Less than 0.1% produce the best outcomes (Match 5, Match 5+ and Match 6) – much too small to observe in the pie chart on the right.
The figures in the table confirm our belief, wrought from experience, that it is difficult to win any prize in Lotto, let alone one of the higher ones, which are the most attractive and desirable.
How to win a prize
The aim is to predict in advance which numbers will be drawn. The minimum entry, called a board, is a selection of six different numbers. There are five categories for each board played, and a winning board receives a prize in the highest category for which the numbers qualify.
| Category | Condition | Example(s) | Possibilities |
|---|---|---|---|
| Match 3 | Any 3 main balls | (3, 5, 14) or (5, 30, 44) | 6C3 = 20 |
| Match 4 | Any 4 main balls | (3, 5, 14, 22) or (5, 22, 30, 44) | 6C4 = 15 |
| Match 5 | Any 5 main balls | (5, 14, 22, 30, 44) | 6C5 = 6 |
| Match 5+ | Any 5 main balls & bonus ball | (5, 14, 22, 30, 44) and 10 | 6C5 = 6 |
| Match 6 | All main balls | (3, 5, 14, 22, 30, 44) | 6C6 = 1 |
You will not be surprised to learn that it is harder to match six main numbers than five, five more than four, and four more than three. Therefore, in general, the cash value of the prizes increases with the number of balls that match, reflecting the degree of difficulty. However, the amount won depends also upon the number of prize winners in each category for a pari-mutuel[2] game. We shall see how this works in another article later.
In a Single Draw
Including the bonus ball, (43 × 49C6) combinations can be drawn. i.e. 601,304,088[3]. There are only six fixed outcomes, regardless of the numbers chosen to comprise a single board. The table below summarises the outcomes.
| ID | Match 3 | Match 4 | Match 5 | Match 5+ | Match 6 | Probability(%) | Occurrence |
|---|---|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 0 | 98.14 | 590,097,256 |
| 2 | 1 | 0 | 0 | 0 | 0 | 1.765 | 10,613,260 |
| 3 | 0 | 1 | 0 | 0 | 0 | 9.686×10-2 | 582,435 |
| 4 | 0 | 0 | 1 | 0 | 0 | 1.802×10-3 | 10,836 |
| 5 | 0 | 0 | 0 | 1 | 0 | 4.291×10-5 | 258 |
| 6 | 0 | 0 | 0 | 0 | 1 | 7.151×10-6 | 43 |
| Totals: | 100% | 601,304,088 |
Only five win a prize, not including the most probable outcome (shown in red). Overall, the probability of winning any prize is 1.86% in any one draw.
Subsequently, the next pertinent question is: How might I fare over an extended period?
2,065 Draws with 49 Balls
To answer the question above, I carried out some experiments covering the period when Lotto was played with 49 balls and then drew a bar chart of the results[4].
The bar chart shows that you are most likely to succeed between 31 and 45 times in 2,065 draws[5], with 38 being the most probable at 6.50%.
Most prizes are likely to be £10 (or later £25, after the New Dawn[6]) with a few Match 4 wins and possibly a Match 5 win, too. Winning a jackpot or Match 5 plus bonus prize is improbable, but not impossible; it becomes a reality only for some lucky players.
Looking at the total cost calculated below, you need to win at least one sizeable Match 5 prize to stand a chance of making a profit.
Total cost over 2,065 draws: (1855 × £1) + (210 × £2) = £2,275
For example, if I had played (4, 7, 9, 23, 28, 37) throughout , I would have won prizes on 46 occasions, collecting a total of £582, but making me a loss of £1,693.
| Category | Prize Breakdown | Amount |
|---|---|---|
| Match 3 | (40 × £10) + (4 × £25) = £400 + £100 | £500 |
| Match 4 | £40 and £42 | £82 |
| Match 5 | None | £0 |
| Match 5+ | None | £0 |
| Match 6 | None | £0 |
| Totals: | £582 |
This stark finding demonstrates how difficult it is to win a life-changing sum of money randomly.
That’s enough for now. We shall look further into this problem in part 2 of this article.
Notes
- [1] I have expressed small percentages in scientific notation to maintain a precision rounded to 3 significant figures rather than use the appropriate number of zeros after the decimal point. e.g. 1.23×10-4 instead of 0.000123%. The effect is tidier.
- [2] Lotto is based on a system of betting known as a pari-mutuel, in which winners receive a proportion of the total money staked after deducting management and other charges, such as tax.
- [3] From now on, we shall account for the bonus ball since it generates outcomes that we ought to consider. However, including the bonus ball becomes significant only when we buy more than one board.
- [4] The Binomial Distribution function allows us to calculate the probability of winning multiple times over several attempts. Fortunately, to simplify things, the Binomial Distribution function is available in Microsoft Excel through the worksheet function BINOM.DIST.
- [5] The width at half-maximum of a peak is a common way of measuring overall success. By this measure, between 31 and 45 wins account for more than 75% of those observed in practice.
- [6] Camelot created a New Dawn for Lotto from 5 October 2013 by making changes to reinvigorate the game. The board price doubled to £2, and the Match 3 prize rose from £10 to £25. Also, raffle prizes of £20,000 and £1 million were added to spice up the higher tiers and arouse interest.
Revision date: 12 February 2023