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cost to buy every powerball combination

How Many Lottery Combinations Do You Need to Guarantee a Jackpot?

There’s a Surefire Way to Win a Jackpot. But Is It Worth It?

You know that the odds of winning a lottery jackpot are sky-high, right? Well, that’s only kind of true. There are a limited number of possible winning numbers, so the odds of hitting a jackpot can be 100% if you buy enough lottery combinations. But how many ticket combinations would you have to buy? How much would you have to invest to do it? And would your investment be worthwhile?

The (Only) Guaranteed Way to Win a Powerball or Mega Millions Jackpot

Lotteries like Powerball or Mega Millions differ from sweepstakes because the winner isn’t randomly drawn from all of the qualified entries. Instead, entrants play by buying a ticket with a combination of numbers, and one of those combinations is drawn as the jackpot winner.

That means that the chances of winning are fixed and that they aren’t influenced by how many people buy tickets. Because there are a fixed number of lottery combinations, a very determined entrant with enough money at their disposal could buy every single possible combination and guarantee a jackpot win.

Take the Powerball lottery, for example. To win the jackpot, you need to have a lottery ticket with the correct combination of five white balls and the red Powerball. There are 69 possible numbers for the white balls and 26 possible results for the Powerball. Thus, the odds of picking that perfect combination with a single ticket are one in 292,201,338.

Each Powerball ticket costs $2. That means you could buy all the possible combinations of tickets for $584,402,676.

You can also guarantee a jackpot win for Mega Millions. For Mega Millions, you need to have a lottery ticket with the correct combination of five white balls and the Mega Ball. The five white balls have possible numbers ranging between 1 and 70, while the Mega Ball can be between 1 and 25. That means you need to cover 302,575,350 combinations to guarantee a jackpot.

Mega Millions tickets cost $2 each so covering all combinations would cost $605,150,700. In some regions, you can also buy a “jackpot only” ticket that covers two combinations for $3. That means that if you want to guarantee a jackpot for the lowest possible investment, you could do it for “only” $453,863,025.

Given that some of the largest lottery jackpots can reach a billion dollars or more, that doesn’t seem like such a bad deal, right? But of course, it’s not that easy.

Does It Make Sense to Buy All Possible Lottery Combinations?

While covering all of the possible combinations takes an astounding of money, some of the biggest lottery jackpots have advertised jackpots that outstrip what you’d need to pay to win them. Powerball’s biggest jackpot to date was advertised at $1.5 billion. Tripling your investment on a guaranteed jackpot sounds pretty good, doesn’t it?

The problem is that, while you can guarantee a jackpot win, you can’t guarantee that you will end up with a profit. Even with an advertised jackpot bigger than the amount you’d have to invest, there are costs that will eat into your earnings.

First of all, you might have to split the jackpot with other winners. In the case of Powerball’s $1.5 billion jackpot, there were three winning tickets. That means even the simple math of $1.5 billion divided by three winning tickets would have brought the value of the prize below the $584,402,676 in ticket combinations you would’ve had to pay to guarantee the win.

But you have to do more than simple math to find out what you will net when you win the lottery. For one thing, you have to pay taxes on those winnings. You can expect to pay at least 25% in federal taxes on your prize, and you might be responsible for state taxes on your jackpot as well, Depending on the state you live in, that could add up to another 8.82% that you need to pay to the government.

Next, you’d have to consider whether you’ll take the lump-sum or annuity payout. You’ll only get the full advertised amount of the jackpot if you take the annuity option, but that means that you will have to wait 30 years until you see your return on investment. There are many other ways to invest half a million dollars that could be more profitable and offer more liquidity.

If you take the lump-sum payout, you’ll receive significantly less money. In the case of the $1.5 billion Powerball jackpot, each of the three winners took the lump sum and received $327.8 million instead of $500 million.

So far, the largest jackpot won by a single person was a $758.7 million dollar Powerball jackpot. Even then, the winner only walked away with $336 million after taxes and the lump-sum reduction, far less than it would have cost to buy all possible lottery combinations.

Finally, there are a number of things that you should do before you cash in a major lottery win, including hiring lawyers and accountants to protect your interests. Hiring good people is important, but it costs money, further eating into your jackpot profits.

Can the Jackpot Get Big Enough to Be Worth Buying All Combinations?

So maybe a $1.5 billion jackpot isn’t enough to be worth buying all of the possible ticket combinations. But jackpots can grow ever larger than that, can’t they?

Well, kind of, but not really. As jackpot values rise, lottery fever kicks in and more and more people buy tickets. The more tickets get sold, the higher the chances that all possible combinations will be covered. So before the jackpots get close to big enough to be worth buying all the tickets, someone will almost certainly buy a winning ticket.

Conclusion

While you can guarantee a lottery jackpot given enough money, it rarely works in your favor. And the chance of a lottery jackpot getting big enough to ensure a good return on your investment is slim. When the $1.5 billion Powerball jackpot was won, lottery fever was so high that 89 percent of all possible combinations had been purchased. It’s highly unlikely that a Powerball jackpot will ever get much higher than that.

There are better strategies to win the lottery than tying up a half a million bucks in lottery tickets. Treat the lottery as it’s intended: a game, not an investment strategy. Buy a single lottery ticket any time the jackpot soars over $350 million, which is the point at which the risk becomes worth the $2. Then cross your fingers, hope for good luck, and have fun with the results.

You can guarantee a jackpot win by buying enough lottery ticket combinations. But how much would it cost to do it, and is the result really worth it?

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