Why It’ll Never Make Sense to Play Powerball (But We’re All Going to Play Anyway)

Would it be worth buying every Powerball number to “guarantee” a win? As you would likely expect, no, not yet, and maybe not ever. With the jackpot currently at $1.5 billion, you’d need a lot more jackpot to make up for the multitude of variables at play in the game.

The Premise

This analysis looks at the ROI or expected value of spending $584,402,676 to purchase every possible combination of Powerball tickets available. Forgetting the practical considerations of raising such funds or actually purchasing the tickets, including recycling the 280,450,800 tickets that are losers, what would it take for this to actually make sense? The result is a combination of expected values from the non-jackpot prizes, the elusive jackpot, cash values, taxes and many other variables at play, which your purchase alone would alter.

The Non-Jackpot Prizes

Let’s first start with calculating out the non-jackpot prizes, a relatively easy task. In most states, they are fixed prizes (California spoils the pot, as they will several times, by using pari-mutuel prizes), and the odds of drawing them are easy to calculate.

Total Prizes

The table below shows the total prizes won if you were to purchase every possible combination. Note that the odds of matching just the Powerball (1-in-26) are actually higher (north of 1-in-38) because, in this case, you are calculating the odds you match just the Powerball and no other numbers. You will match the Powerball 1-in-26 times, but for a portion of those, you will also hit other numbers and win bigger prizes (like, you know, the jackpot).

Matching Numbers Number of Wins Prize per Win Total Winnings
5 25 $1,000,000 $25,000,000
4+1 320 50,000 16,000,000
4 8000 100 800,000
3+1 20,160 100 2,016,000
3 504,000 7 3,528,000
2+1 416,640 7 2,916,480
1+1 3,176,880 4 12,707,520
0+1 7,624,512 4 30,498,048
Grand Total $93,466,048

(All that attention to the 5+0 million-dollar prize? You actually make more on all the simple Powerballs you will match than those more difficult matches.)

Power Play Option

But wait! What about the Power Play? Not so fast.

Powerball also lets players make a side bet on the non-jackpot prizes. For an extra dollar per play, you can multiply your non-jackpot wins by anywhere from 2x to 5x. (There is an option for a 10x multiplier, but that ball is not in play for jackpots over $150 million, thus we disregard it.)

Powerball determines the specific multiplier using a separate drawing using 42 balls. The allocation of these balls weighs toward the lower end, with 24-in-42 drawing the 2x multiplier, 13-in-42 drawing 3x, 3-in-42 drawing 4x and 2-in-42 drawing 5x.

Although the Power Play increases the total winnings, the increase does not justify the increased cost ($292,201,338 for all the extra $1 plays). All told, the expected value of the non-jackpot prizes increases to $179,685,696 with the Power Play option. This is well above the $93,466,048 you would expect without it, but the increase is still $205,981,690 short of making up for the total extra Power Play cost. (In other words, a sucker side-bet; 50 points to California for not allowing this play.) Therefore, for the purposes of this analysis, we will exclude the costly Power Play option.

Taxes

You are going to have to pay them anyway, so for now and while we’re here, let’s factor in taxes on the non-jackpot prizes (which will actually make it easier to factor in taxes on the jackpot prize later on).

For federal taxes, the federal government uses graduated tax brackets which increase a person’s tax percentage as their income increases. However, contrary to some belief, these increased tax rates apply only to the money made after hitting a new bracket. In other words, the IRS taxes the first $9,275 someone makes at 10%; any money beyond that is taxed at the increasing graduated rates.

The top federal tax bracket for 2016 is 39.6% for income over $415,050. This means that for the first $415,050 you win, the IRS will tax you at the sum of the lower tax brackets (that sum works out to $120,529.75 for the first $415,050, an effective tax rate of about 29%). Any income above and beyond $415,050 (like the billions you hope to win) will be taxed at 39.6%. For the purposes of this analysis, we’ll take advantage of the lower tax brackets while calculating out the non-jackpot prizes’ taxes, which will allow us to simply tax the jackpot prize at a flat 39.6%.

For state taxes, these vary widely, but some states either don’t have a personal income tax or don’t tax lottery winnings (look to FL, NH, SD, TN, TX, WA or WY for these state-tax havens; CA and PA don’t exempt lottery tickets bought in those states). For the purposes of this analysis (and since I’m in exempt California; another 50 points awarded there), we’ll neglect to calculate in state income taxes. Furthermore, if you’re going to drop $584 million on lottery tickets, you might as well do it in a tax-friendly state.

To calculate the taxes on the non-jackpot winnings, we first knock off $415,050 from the winnings and tax those at the summed-up graduated figure of $120,529.75. This leaves $93,050,998 subject to the 39.6% tax rate, or $36,848,195.21 in taxes. Your total taxes on the non-jackpot winnings will then be $36,968,724.96.

Therefore, for the non-jackpot winnings, you can expect to make $56,497,323.04 after taxes. Nice chunk of change, but you still have $527,905,352.96 to go to make this deal worth it.

The Jackpot

The jackpot, of course, is more difficult to factor in. This is because the jackpot is subject to two major variables:

  1. The jackpot’s cash value (notwithstanding its announced, annuity-based value), and
  2. How many people win with you.

Furthermore, the jackpot is set to increase as tickets sales increase (about 34% of each ticket cost goes into the jackpot’s current cash value).This affects the jackpot, especially given the nearly $200 million you’re about to dump into it.

Jackpot Cash Value & Taxes

As most people are aware, the jackpot’s announced value (currently $1.5 billion) is actually the jackpot’s future value after investment. This option is paid out in 30 gradually-increasing payments (to allow for inflation) over the next 29 years (the first payment occurs upon claim of a winning ticket). The announced value presumes how much the jackpot’s current cash value would earn on government-backed securities over the next 29 years if those securities were purchased upon winning.

Players can select to just receive this cash value. In practice, the jackpot’s current cash value averages out to about 62% of the announced jackpot’s value. Thus, for the purposes of this analysis, we figure you will earn 62% of the jackpot before taxes.

Since we’re here, let’s factor in taxes now so we can set our jackpot target amount. With taxes of 39.6%, you will keep 60.4% of the jackpot after taxes. Thus, we can multiply 62% times 60.4% to calculate the current, after-taxes jackpot amount you’d keep. This figure is 37.448%.

But before we can factor this into our needed amount, we owe ourselves lot of credit. Even though 280,450,800 of our tickets will be losers, money from those purchases (and, in fact, all our ticket purchases) will go right back into the jackpot, which we’re guaranteed to win since we already bought every ticket. Powerball puts about 34% of each ticket’s sales into the jackpot’s current cash value, so we’ll essentially get that money back.

Thus, we won’t need to get to our $527,905,352.96 current-value, after-taxes goal because we’ll be getting $198,696,909.84 back from our ticket purchases alone. Thus, to fully meet our needs to get our money back, we need to have a jackpot with an announced value of at least $879,108,221.71, since our ticket purchase and its corresponding cash infusion into the jackpot will jump us up to the $1,409,702,395.21 we’ll eventually need to cover our losses. (Most folks forget to consider this consideration when buying each ticket.)

Chances You Split the Jackpot

Now, if the lottery were nice and let anyone who matched all 6 numbers win a jackpot of $879 million or more, we’d be done with the analysis. But no! This is a cruel world that will make you share your winnings.

As folks know, if multiple people match all 6 numbers, they both win the jackpot, but that jackpot is then equally split among those winners.

Now, if only two people play, the odds of both those people winning is extremely remote (0.00000000000000117% chance). But if three people play, the odds of at least one (and maybe more than one) person winning improve ever so slightly. But what if 4 people play? 400? 400 million? As you can imagine, the odds of more than one person winning increase. There is therefore a correlation between the increase in tickets sold and the increase in the likelihood of more people winning.

Factoring the Likelihood

So how do we figure out how many people will likely win the jackpot? We can start by figuring out the odds that no one wins the jackpot and work backward from there.

As an example, take a fair coin (heads or tails; 50% likelihood of each occurrence happening). If you flip the coin once, what are the odds it will not land on heads? Clearly, since it’s a fair coin, 50%.

But what if you flip the coin twice; what are the odds that it will not land on heads for either toss. The answer here is 25%; 50% for the first flip and, if that happens, 50% for the second. What about for three flips? 12.5%; or 50% times 50% times 50%. The formula here is essentialy (1-p)^n, where p is the probability of the thing you’re trying to avoid (like flipping heads or, later on, matching all 6 numbers) and n is the number of times you flip the coin (or, for later on, how many tickets are sold).

Essentially, what we’ll do is drop the probability of “hitting heads” (or hitting the jackpot) from 50% to 0.000000342%, or 1-in-292,201,338 odds. We’ll also have to increase the number of “flips” (or tickets sold) to… wait, what number do we increase this to?

Calculating the Tickets Sold

[DISCLAIMER: this is admittedly the trickiest math to compute, and I’m not 100% sure the calculation below is correct. I certainly welcome thoughts and differing ways to calculate this figure.]

If we can figure out how many tickets will be sold for the jackpot drawing, we can then figure out the chance that no one wins the jackpot.

A website does compile the ticket sales data for Powerball and its competitor, Mega Millions. From here, we can compare the jackpot announced prize and see the corresponding number of tickets sold for that drawing. As you can imagine, as the jackpot’s announced value increases, so do ticket sales. This is a two-way street; an increased jackpot increases demand (more people want to win), and these increased sales increase the jackpot (because, as mentioned above, about 34% of ticket sales go to the jackpot’s current cash value, which, in turn, increases the jackpot’s announced value).

The chart below shows a scatterplot of the jackpot-vs.-sales data, and it is noticeably parabolic in shape. Thus, as the jackpot’s announced value increases, ticket sales exponentially increase as well. (This chart and its data includes all draws since Powerball increased ticket sales to $2 per draw.)

Powerball Jackpots vs. Number of Tickets Sold with regression line

This chart also gives us a regression line to calculate the number of tickets sold for any given announced jackpot value. We’ll use that regression formula to compute out our tickets sold.

Danger: Faulty Data Ahead

However, we’re now entering a danger zone with respect to statistics, as the data we are seeking is less and less reliable. This is because we are now essentially dealing with (and calculating out) outliers, or data which “lies outside” the bulk of the data’s sampling. For example, you can see that people were really excited for that first $400+ million jackpot, buying nearly 300,000,000 tickets. However, future $400+ million jackpots did not fuel such demand, and that figure was only topped with last Saturday’s near-billion-dollar jackpot. (As a show for how outlying the first $400+ million jackpot data point is, excluding that jackpot from the data set raises the regression line’s statistical confidence from 89%, as noted above, to over 96%.)

It is also likely that the regression line skews upward, disproportionately exaggerating ticket sales, which is exactly where we are trying to calculate with an announced jackpot value of at least $1.5 billion. You can note how the regression line already starts to curve above our last data point. It is also likely that at some point, ticket sales will plateau as the lottery-purchasing public will have their demand saturated.

[One commenter also correctly points out that the purchase of all tickets would have to be in secret; the public knowing about someone having every ticket would presumably chill ticket sales drastically.]

Using the regression formula, we can calculate out our presumption for how many tickets will be sold for any given announced jackpot value. However, this is where things fall of the hinges. With the current announced jackpot value of $1.5 billion, the regression formula predicts that 1,178,360,650 tickets would be sold, shattering the record from Saturday of about 440 million. It is frankly implausible (though still possible) that ticket sales will reach that level (this alone would add about $800 million to the jackpot, given 34% of the money from tickets sold goes to the jackpot).

With nearly 1.2 billion tickets sold, there would be a 1.77% chance that no one wins the jackpot tomorrow. This would also mean that (as described below), there would be a 50% chance that 39 people win the jackpot, dividing it that many ways. Such a projection would deflate the jackpot’s expected value (taking the sum of the probabilities of possible number of winners times the corresponding jackpot split) to about $108 million, well short of our need of about $879 million.

To make up for this, we would need to increase the announced jackpot value over $1.5 billion; however, this also sends us on an upward, exponential spiral where the increasing jackpot spurs more and more ticket sales, which requires a bigger and bigger jackpot amount, which spurs more and more… Dizzy yet? (Plus, as we noted above, our formula may be way off, too.)

Instead, we’ll have to assign a presumed ticket sale level based upon our best presumptions for how many tickets will sell. Admittedly, this takes us outside the realm of statistical calculation and into the presumptions we were trying to avoid in the first place. Yet to show these figures, let’s look at some historical examples and some projections.

More Plausible Ticket Sales Data

Saturday’s drawing spurred 440, 321,172 tickets sold, a record for Powerball. Let’s use this as a starting point for our number-of-winners odds. Using our formula from above, (1-p)^n, we can calculate the odds that no one will win the jackpot. This figure is 22.16%; in other words, there was a 77.84% someone should have one last Saturday’s jackpot, but no one did.

With 22.16% as our staring point, we presume (perhaps incorrectly) that that same calculation applies for the remaining 77.84% likelihood that someone won the jackpot. In other words, out of the 100% chance that all 440,321,172 tickets are winners (hey, it could happen), we’ll cut off 22.16% of that pie and use the remainder to calculate the odds that only one person wins the jackpot. We’ll then keep sectioning off pieces of the pie until we reach 440,321,172 total winners (or, practically, until the odds of such high splits are negligible).

Here, we basically redo the calculation using the 22.16%, because at this point, we’re already presuming that someone wins. The 22.16% is still a fair number, although our basis for that derivation now needs to presume that there are only 440,321,171 tickets left to test (because we already had that one winner, and we’re testing the odds that none of the other tickets win). Given the negligible effect this small change would have on the 22.16% figure, we can simply excuse such variance.

We then cascade the proportions forward from no one winning, to exactly one person winning, to exactly two people winning, to exactly three people winning, and so on. At some point, the likelihood that multiple people will win diminishes to zero, similar to Zeno’s Paradox.

Therefore, we can presume that 22.16% of the times you will win alone, 17.24% of the time one other person will win, and so on. We can also make this presumption because their winning is independent of your ticket purchases. The table below shows the percentages given varying ticket sales levels, starting with last Saturday’s figure.

# of Other Winners ~440 million 550 million 650 million 750 million
0 22.16% 15.22% 10.81% 7.68%
1 17.25% 12.91% 9.64% 7.09%
2 13.43% 10.94% 8.60% 6.54%
3 10.45% 9.28% 7.67% 6.04%
4 8.14% 7.86% 6.84% 5.58%
5 6.33% 6.67% 6.10% 5.15%

To wind up our calculation, let’s just very arbitrarily, very unscientifically presume that tomorrow’s jackpot will include sales of 750 million tickets. Is this a fair number? Perhaps, but we honestly don’t know because we’ve never been here before. It is possible that the scatterplot will flatten at some point, and whether that is for a jackpot between $950 million and $1.5 billion will be seen tomorrow. However, we’ve also seen record-breaking jackpots foster outlying data in ticket sales, so maybe we’re still lowballing tomorrow’s sales figures.

Based upon these figures, and continuing down the line of numbers until they are statistically negligible, we can calculate out the expected value for each number of total winners, using as our base presumption that there is a 7.68% chance that no one else wins.

Summing up the expected values, it would take a jackpot of (drum roll, please) $4,117,931,309.58 to yield an expected return of the $879,108,211.71 we’ll need to break even. (For note, the regression formula predicts that over 8.75 billion tickets would be sold if the jackpot were ever that high.)

And that’s the best part: even with this jackpot, you walk away breaking even (technically, you earn a fraction of a cent, but who’s counting). But, the good news is, for any jackpot higher than our $4.1 billion figure, you’re (maybe) in the black and (maybe) making pure profit. If the jackpot were $5 billion, for example, you’d (expect to) walk away with $70 million, cash money (the remaining variable will still be the actual number of people you split the jackpot with).

How do we get to that $4.1 billion number? The table below shows a snippet of the data given that announced jackpot value and 750 million tickets sold.

Number of Other Winners Likelihood of Result Expected Value
0 7.68% $316,202,509.06
1 7.09% 145,961,175.05
2 6.54% 89,835,528.45
3 6.04% 62,203,013.50
4 5.58% 45,941,317.64
5 5.15% 35,344,694.80
6 4.75% 27,969,163.73
7 4.39% 22,593,815.04
8 4.05% 18,541,253.07
9 3.74% 15,405,777.67
10 3.45% 12,929,834.79
11 3.19% 10,942,245.38
12 2.94% 9,324,947.14
13 2.72% 7,993,992.38
14 2.51% 6,888,149.14
15 2.32% 5,961,778.72
16 2.14% 5,180,228.88
17 1.97% 4,516,764.01
18 1.82% 3,950,466.10
19 1.68% 3,464,766.56
20 1.55% 3,046,398.52
242 0.00% 0.01

As shown, you would expect to earn about $316 million from the expectation that you win alone, about $146 million that you win with only one other person, $90 million that you win with two other people and so on. Because these are expected values, remember, you sum up the expected values of all the probabilities to arrive at your total expected value (our $879 million target). For these figures, that sum went all the way to 242 people winning. Given these numbers, we expect you to split the jackpot with at least 8 other people (51.28% chance of that happening).

Conclusion

So, is it worth spending $584,402,676 to buy every Powerball ticket? Maybe when the jackpot hits well north of $4.1 billion, and, only then, if ticket sales finally flatten out.

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