## 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.

#### Single 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.

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

## We can think of 3 major problems with buying 292,201,338 lottery tickets with every combination of Powerball numbers

In a Powerball draw, five white balls are drawn from a drum with 69 balls and one red ball is drawn from a drum with 26 balls. If you match all six numbers, you win the jackpot. If you partially match some of the numbers, you win a smaller fixed prize.

There are 11,238,513 ways to draw five white balls from a drum of 69 balls. Multiply that by the 26 red balls, and there are a total of 292,201,338 possible Powerball tickets.

At $2 for each ticket, then, it would be possible to buy every possible ticket for $584,402,676. As a journalist, I don’t have that much money sitting around, but either a consortium of a few million Americans or a large and wealthy institution like a bank could conceivably assemble that level of cash.

With the sky-high jackpot in play, this actually at first glance guarantees a profit — at least before taxes. Since we’ve bought every ticket exactly once, we can see how much we will win based on the jackpot and the smaller prizes:

Indeed, this is something of a low-ball estimate. As we are buying another half-billion dollars’ worth of tickets, part of that money will be added into the jackpot pool.

Of course, there are a few extra complications to this project.

### Actually buying 292 million tickets

The first problem is the actual physical act of buying 292 million Powerball tickets and filling them out by hand. Since we need to very carefully and systematically make sure we get every possible ticket, using the computer-generated random quick draw will not work for us.

According to Statista, JPMorgan Chase Bank has about 189,000 employees. That means that there are about 1,546 possible Powerball tickets for each employee. If each employee spent 10 hours a day buying and filling out Powerball tickets for three days, this would mean each employee would need to fill out about 50 tickets per hour. So while this would be extremely difficult to do and perhaps not the best use of a large organization’s resources, it seems that it might be physically possible, if somewhat grueling, to actually buy every Powerball ticket.

Similarly, a large, decentralized consortium of several thousand or a few million Americans connected over the internet — something like an office Powerball pool on a mass scale — would be physically capable of buying 292 million lottery tickets. Of course, the logistical coordination of such a consortium would be a daunting task, and one could imagine various legal and practical difficulties with distributing the money after the drawing.

### Splitting the jackpot

The second and larger problem with our comprehensive Powerball scheme is the risk of splitting the jackpot. While the fixed prizes do provide about $93 million of our winnings, the overwhelming bulk of the money comes from the big prize.

That would mean splitting the jackpot two or more ways with other players would be absolutely devastating to our plan. A two-way split cash-prize jackpot would give us $465 million before taxes. Adding in the fixed prizes, we get a total of about $558 million in winnings, which is now less than the ticket costs of about $584 million, leaving us a loss of nearly $26 million.

The likelihood of splitting the pot is determined by how many other tickets are sold. Business Insider looked at this after the January 6 drawing in which there were no winners, paving the way to the current insanely high jackpot. Following the logic from that post, we can estimate our odds of getting the jackpot alone based on a few guesses about ticket sales.

According to LottoReport.com, a site that tracks lottery sales and jackpots, 440,321,172 tickets were sold before Saturday’s drawing. With that many tickets sold, and under the assumption that everyone else playing Powerball is picking numbers more or less at random and independently from each other, there’s just a 22% chance that we would be the only winner.

We could also expect that, with the over a billion-dollar headline prize, even more tickets will be sold before Wednesday’s drawing, greatly hurting our chances of walking away with the full jackpot without having to share:

### Other people trying the same thing we are

The above analysis of our odds of splitting the pot assumed that all the other tickets sold were to normal people who would choose their numbers more or less at random. But seeing as we are going all in and buying every ticket, it’s possible that someone else could be attempting this as well. There are, after all, several organizations in the US that have the financial and personnel resources to theoretically go out and buy 292 million Powerball tickets.

Of course, if two or more banks or consortia tried this plan, they would be certain to have to split the pot and thus lose a bunch of money. This situation is similar to the game Chicken, in which two drivers start out driving directly at each other. If one driver swerves while the other keeps going straight, the first driver “loses” and the second driver “wins.” If both drivers swerve, the game is a draw. Naturally, if both drivers keep going straight, their cars crash and they die in a fiery wreck.

In Chicken, the strategy you adopt depends on what you think the other driver is going to do — assuming you’re actually playing something as reckless and stupid as Chicken in the first place. If you think he’s crazy enough to keep barreling forward, you should be more likely to swerve. If you believe, on the other hand, that he’s going to veer out of the way first, then you might be more likely to keep driving straight.

Banks or billionaires with thousands of employees that are considering buying every Powerball ticket need to make a similar consideration. If there’s a low likelihood that a competitor is going to also mobilize a small army of people in a bid to win a historically high lottery jackpot, then perhaps that risk is worth taking. If, on the other hand, we think that there might be not just one but several other wealthy organizations or people that are making similar plans to our own, we should stay out of the fray.

On the one hand, you would definitely win the jackpot. On the other, you'd probably have to share it.