Put away your crystal balls and stop looking up your cats birthdays. There is, theoretically, a way to guarantee you win the lottery jackpot and it’s actually quite simple – buy every number combination. But what would happen if you did? If the prize fund were large enough, would you make a profit?

Please Note: This article is for entertainment purposes only, and is intended as an exercise in mathematics and probability. We aren’t saying you should actually attempt to buy 14 million lottery tickets.

How Many Number Combinations Are There?

For this exercise we are going to use the UK National Lottery (Lotto) in which players select 6 numbers between 1 and 49 (inclusive). Six regular play numbers are drawn, along with a seventh ‘bonus ball’ number.

Working out the number of unique combinations is relatively simple. You have 49 choices for your first number, 48 for your second, 47 for your third and so on. In the UK National Lottery (Lotto) the order in which the winning balls are drawn does not matter, for this reason we also need to divide the number of choices by the number of available positions – so 49/6, 48/5, 47/4 etc…

The number of possible combinations is therefore:

49/6 x 48/5 x 47/4 x 46/3 x 45/2 x 44/1 = 13,983,816 (approximately 14 million).

What Are The Odds Of Winning A Prize?

I am not going to go into calculating the odds of each specific prize here. If you are interested, we discuss how to calculate the odds of winning a specific prize in the National Lottery (Lotto) in more detail in the article: National Lottery Odds – What Are The Chances Of Winning The Lotto Jackpot?

Three Numbers: 1 in 56.7

1 in 56.7 Four Numbers: 1 in 1032

1 in 1032 Five Numbers: 1 in 55,491

1 in 55,491 Five Numbers + Bonus Ball: 1 in 2,330,636

1 in 2,330,636 The Jackpot – Six Numbers: 1 in 13,983,816

What Is The Prize Money & How Is It Divided?

Buying all 13,983,816 ticket combinations would cost you £13,983,816. But how much would you win? This depends on the size of the prize fund, which is directly related to how many tickets have been purchased, and whether or not it is a rollover.

How Big Is The Prize Fund?

For every £1 ticket purchased, approximately 45p goes into the prize fund, whilst the other 55p goes to the various charitable “good causes” with a percentage being held back for operational costs, retail costs and profit.

What Is The Prize Fund Distribution?

Matching 3 numbers pays a fixed £10. All other prizes are paid as a percentage of the remaining prize fund distributed equally between the all of the players that have won that specific prize. For example, if the allocated prize money for 5 numbers was £100,000 and 10 players matched 5 numbers, each would receive £10,000.

The percentage of the prize fund allocated to each prize category is as follows:

Three Numbers: £10 (fixed)

£10 (fixed) Four Numbers: 22%

22% Five Numbers: 10%

10% Five Numbers + Bonus Ball: 16%

16% The Jackpot – Six Numbers: 52%

If I Bought All The Combinations, How Much Would I Win/Lose?

How much you would actually win would vary, depending on how many other winning tickets there were. But we can use the maths to work out the theoretical winnings.

The average prize fund, according to the National Lottery (Lotto) is approximately £4 million (~8,888,889 tickets) – this is before we have bought any tickets. If we add the prize fund contributions from our ticket purchases, this would become £10,292,717.20 (Original £4m plus £13,983,816 in ticket purchases x 45%). The total number of tickets in the game would be approximately 22,872,705.

Prize Fund Distribution

Using the numbers above gives us the following prize distribution data:

Prize Chance Of Winning ^ Prize Allocation (%) Prize Allocation (£) ^ Number Of Winners Prize Per Winner ^ Jackpot 1 in 13,983,816 52% £3,252,908 1.64 * £1,998,749 Five Numbers + Bonus 1 in 2,330,636 16% £1,000,895 9.81 * £101,987 Five Numbers 1 in 55,491 10% £625,559 412 ^ £1,518 Four Numbers 1 in 1032 22% £1,376,230 22,155 ^ £62 Three Numbers 1 in 56.7 £10 fixed £4,037,125 403,712 ^ £10

* To 3 significant figures. ^ To 0 decimal places