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What Happens if you Buy Every Combination of Lottery Tickets?

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 
  • Four Numbers: 1 in 1032
  • Five Numbers: 1 in 55,491
  • Five Numbers + Bonus Ball: 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)
  • Four Numbers: 22%
  • Five Numbers: 10%
  • Five Numbers + Bonus Ball: 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

    How Much Do We Win?

    We can use the data to estimate our theoretical winnings:

    Prize Number Of Wins Prize Per Winner ^ Win Per Prize Category ^
    Jackpot 1 £1,998,749 £1,998,749
    Five Numbers + Bonus 6 £101,987 £611,923
    Five Numbers 252 £1,518 £382,452
    Four Numbers 13545 £62 £841,394
    Three Numbers 246820 £10 £2,468,200
        Total Win:  £6,292,717

    ^ To 0 decimal places

    So our total winnings are £6,292,717 which is 45% of our original purchase - the same percentage that is allocated to the prize fund. In this example our winnings (£6,292,717) minus our ticket costs (£13,983,816) would result in a net loss of £7,691,099.

    What Happens When There Is A Rollover?

    As you saw in the above example, it is not possible to make a theoretical profit in a regular National Lottery (Lotto) draw. In fact, buying every ticket results in a substantial loss of 55%. What what about a rollover draw?

    A rollover occurs when no one wins the jackpot in a given draw. The jackpot prize money is then "rolled over" and added to the next weeks jackpot. This can happen up to three times (rollover, double rollover and triple rollover). Having a rolled over jackpot changes the game significantly as the prize fund increases in proportion to the number of tickets purchased. The effect is watered down somewhat by the increased number of players, but is it enough to guarantee you a theoretical profit?

    In this next set of examples, the number of players in a draw affects the rolled over jackpot amount. For this reason we will use the real data from a rollover, double rollover and triple rollover that occurred in April 2010.

    Rollover

    For the first rollover we have used the prize fund data from National Lottery Draw #1491 held on Wed 7th April 2010. The draw included a rolled over jackpot of £4,194,487 which was added to a prize fund of £8,773,227. Approximately 19.4 million tickets were purchased. To this draw we have added our fictional purchase of 13,983,816 tickets, and our prize fund contribution of £6,292,717.

    Prize Prize Allocation (%) Prize Allocation (£) ^ Number Of Winners Prize Per Winner ^ Our Share
    Jackpot 52% + Previous Jackpot £8,943,283 2.39 * £3,745,359 £3,745,359
    Five Numbers + Bonus 16% £1,461,168 14.3 * £101,987 £611,923
    Five Numbers 10% £913,230 602 ^ £1,518 £382,452
    Four Numbers 22% £2,009,106 32,343 ^ £62 £841,394
    Three Numbers £10 fixed £5,893,644 589,364 ^ £10 £2,468,200
            Total Win:  £8,049,327

    * To 3 significant figures. ^ To 0 decimal places

    Adding the rolled over jackpot has increased our theoretical returns by almost £2 million to £8,049,327, and reduced the net loss to £5,934,489.

    Double Rollover

    For the double rollover we have used the prize fund data from National Lottery Draw #1492 held on Sat 10th April 2010. The draw included a rolled over jackpot of £7,058,491 which was added to a prize fund of £16,713,988. Approximately 37 million regular tickets were purchased. To this draw we have added our fictional purchase of 13,983,816 tickets, and our prize fund contribution of £6,292,717.

    Prize Prize Allocation (%) Prize Allocation (£) ^ Number Of Winners Prize Per Winner ^ Our Share
    Jackpot 52% + Previous Jackpot £14,329,525 3.66 * £3,919,367 £3,919,367
    Five Numbers + Bonus 16% £2,237,241 21.9 * £101,987 £611,923
    Five Numbers 10% £1,398,276 921 ^ £1,518 £382,452
    Four Numbers 22% £3,076,207 49,522 ^ £62 £841,394
    Three Numbers £10 fixed £9,023,948 902,395 ^ £10 £2,468,200
            Total Win: £8,223,332

    * To 3 significant figures. ^ To 0 decimal places

    Adding the rolled over jackpot has increased our theoretical returns to £8,223,332, and reduced the net loss to £5,760,484.

    Triple Rollover

    For the triple rollover we have used the prize fund data from National Lottery Draw #1493 held on Wed 14th April 2010. The draw included a rolled over jackpot of £12,084,100 which was added to a prize fund of £14,977,718. Approximately 33.3 million regular tickets were purchased. To this draw we have added our fictional purchase of 13,983,816 tickets, and our prize fund contribution of £6,292,717.


    Prize Prize Allocation (%) Prize Allocation (£) ^ Number Of Winners Prize Per Winner ^ Our Share
    Jackpot 52% + Previous Jackpot £18,806,403 3.38 * £5,563,750 £5,563,750
    Five Numbers + Bonus 16% £2,068,401 20.3 * £101,987 £611,923
    Five Numbers 10% £1,292,751 852 ^ £1,518 £382,452
    Four Numbers 22% £2,844,052 45,784 ^ £62 £841,394
    Three Numbers £10 fixed £8,342,928 834,293 ^ £10 £2,468,200
            Total Win: £9,867,718

    * To 3 significant figures. ^ To 0 decimal places

    Adding the double rolled over jackpot has increased our theoretical returns to £9,867,718, and reduced the net loss to £4,116,098.

    Did We Win?

    The short answer is no. Whilst the addition of rollover jackpots reduced the net loss, the gain was not sufficient to generate a theoretical profit. In time, with enough rollovers, a profit might emerge, however as jackpots are limited to roll over three times, this simply would not happen.