Mathematical Musing: Will I be Buying a Powerball Ticket?

Correction: The cost of purchasing every possible ticket combination was miscalculated in the previous version of this post. It has been changed to the correct value.

Every so often, the news media becomes all abuzz when a particular lottery jackpot starts to grow really large.  Right now is one of those times, with no winner on Saturday putting the jackpot for Wednesday’s drawing at around $1.3 Billion, the largest lottery jackpot in US History.

My students sometimes ask me, as a math teacher and a guy who “knows numbers,” whether I play the lottery. Usually I just smile and tell them I buy the occasional scratch ticket for the fun of it, but almost never anything beyond that. It would require a “special occasion” or a “huge jackpot” for me to consider buying one.

This certainly seems like one of those special occasions.

To understand how to approach this question from a math standpoint, we first need to understand the probability of winning.

Never tell me the odds!

powerball

A standard Powerball lottery ticket

The Powerball is a multi-state lottery game run in 44 states as well as Washington, DC; Puerto Rico; and the US Virgin Islands. Only Alabama, Alaska, Hawaii, Mississippi, Nevada, and Utah don’t run the lottery in their state.

Purchasing a $2 ticket involves choosing six numbers, five from a set of 1-69 and 1 from a set of 1-26 (called the Powerball).  Only if all six of your numbers come up will you be able to win the full Jackpot prize, though there are lesser prizes for matching smaller quantities of numbers. Helpfully, Powerball has a list of prizes and odds on their website, but let’s take a moment to understand how they are calculated

On the high end, the jackpot requires your five numbers, plus the Powerball, to match those selected on the drawing.  Assuming that the drawing is done completely at random, then every possible combination of these 5+1 numbers have the same chance of being chosen. So how many combinations are there?

For the first number you pick, there are 69 choices. You can’t pick the same number again, so there are 68 choices for your second number, then 67 choices for the third, 66 choices for the second, and 65 choices for the last. This would suggest that there are 69*68*67*66*65 = 1,348,621,560 ways to pick your first five numbers, and that gets multiplied by the 26 choices for the Powerball to give an overall number of possibilities of 35,064,160,560.

But that number isn’t correct. The calculation above is known as a permutation, which is a way of counting outcomes assuming that different orders of the same numbers are considered different. Permutations are useful when analyzing how many batting lineups of nine players from a team of 37 are possible, or when asking how many ways you can arrange your five family photographs on a shelf. But that’s not what we need here, because the order that the numbers are selected in is irrelevant. What we need to use is a combination, which counts outcomes in a similar way, but treats every different arrangement of the same five numbers as the same outcome.

To get a combination, we divide the 1,348,621,560 figure above by 120, the number of ways to arrange five numbers. This gives a total number of outcomes for the first five numbers as 11,238,513, and a total number of possible tickets including the Powerball as 292,201,338.  So the probability that you’ll win jackpot is 1 in 292,201,338, which you’ll notice is the exact probability listed on their website.  This probability is ridiculously small.  Play 50 tickets a day every day, and this probability suggests you will still only win once every 16,000 years (and it’s actually worse than that, since the Powerball drawings only happen twice a week).

Incidentally, this also means that buying all possible ticket combinations – and therefore guaranteeing that you have the winner – would cost $584,402,676. In 1992, an Australian investment firm attempted to do this in the Virginia state lottery, buying 5 million of the possible 7 million combinations. This would still be a bad play, however. You might need to split the pot with another winner or winners, and lottery winnings are federally taxed by up to 25%.

But the Jackpot isn’t the only prize. Look at the bottom end. You win $4 if you only match the red Powerball, which as we said you have a 1 in 26 chance of doing. So why are the odds listed as 1 in 38.32? As the FAQ say, that figure is not just the probability of matching the red Powerball, but the probability of matching the red Powerball and none of the other numbers.  Match at least one other number and your payout is different.  To completely miss all five other numbers is a 7624512/11238513, or about 68% chance. Multiply that probability by the 1/26 chance of matching the Powerball, and you’ll get the 1 in 38.32 probability they have listed there.

What did you expect?

Now that we have a sense for the probability winning, what do we do that information? How can we break this down and understand the statistical payoff for playing this game?

Consider again buying every possible ticket.  One of them is guaranteed to be the jackpot, which right now is projected to be $1.3 Billion.  Another 25 of them will have all five numbers match, but not the Powerball, winning the $1,000,000 second place prize.  Increasingly more will win the lower prizes, with 7,624,512 having the correct Powerball, but none of the other numbers matching, to win the lowest prize (and, for what its worth, more than 280 million worthless scraps of paper)

If we took the total amount of money won across all winning tickets, subtracted the $2 cost for all 292 million tickets purchased, and divided by the number of tickets purchased, we’d get an average payout per ticket bought.  This average is called the expected value and is a reasonably good measure of the value of playing a game (ignoring the business about sharing jackpots and taxes and all of that).  We can more conveniently calculate the expected value by merely multiplying each outcome by its respective probability.  The table below shows just that, also adjusting each prize for the $2 cost for the ticket.

Powerball 1

The expected value of the Powerball jackpot

This looks great! The expected value per ticket is $2.77, suggesting a positive outcome.  But remember, we’re ignoring a lot here, taxes and the possibility of splitting the pot.  One thing not even mentioned: if you want all $1.3 billion, you will have to wait over a period of 30 years, as the jackpot is only paid out over the long term.  If you want the whole amount as cash up front, the actual payout is $806 million.  How does that affect the expected value?

Powerball 2.PNG

Expected value of selecting the lump sum, cash up front value

The expected value is still positive, but now less than half of what it was before.  But this still isn’t taking taxes into consideration.  USA Mega Jackpot Analysis breaks down tax rules for lottery winnings state-by-state, and New York has the highest state tax rate of any other state at 8.82%.  It projects the after-tax total take-home amount of the lump-sum payout to be $533,410,800.  All the other lesser prizes will be similarly affected.  How does that affect our expected value?

Powerball 3

Expected value of lump sum prize value after tax

As you can see, even after taking the lesser lump-sum value of the prize instead of the higher long-term payout, and after factoring in taxes, the expected value is still positive (though just barely).

So is it worth it?

In short, yes.  For perhaps the first time in the history of the lottery, a single-winner would statistically expect a positive return on their “investment”

Then again, If you have to split the pot, the top jackpot value will get cut into equally sized pieces, which will send the expected value into the negatives (-$0.85 if split two ways, -$1.15 if split three ways, -$1.30 if split four ways).  On the other hand, the value of the jackpot does depend on how many tickets have been bought, and with all the media coverage this record-breaking jackpot is getting, it wouldn’t be surprising to see the jackpot push closer to $1.4 billion by Wednesday.

I’ll see you in line for a ticket.

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7 thoughts on “Mathematical Musing: Will I be Buying a Powerball Ticket?

  1. Are there reasons that people don’t pool together money and buy all the tickets, like the Australian investment firm, other than the sheer amount of money needed to do so? If there is a definite profit, can’t the buyers just split up the winnings proportionate to the amount they contributed to buy all the tickets and then everyone will make a profit over the payout period?

    Very interesting read.

    • There was another story I read once where this actually happened, someplace in the Midwest I think, but I couldn’t find information about it. The main problem is how long it would take to process all of those ticket orders, which was why the firm didn’t actually manage to get all 7 million combinations. That, and if even one other winning ticket was sold to someone else, the group would lose money on the scheme.

      • I see, thanks. It still seems very unlikely to have another winner get the jackpot numbers on the same exact day (if someone were to try to buy all the combinations of numbers and get all of them processed), but if this were to happen, would half of the payout be given to each ticket owner?

  2. That was an awesome read! Do you know for a fact if the numbers drawn are truly random? Is there a history that may possibily reveal previous outcomes are not 100% independent of future outcomes?

    • Also, after looking at the white ball frequencies, 16 and 33 have been picked 5 times. Would it be to your benefit to pick numbers that have appeared more often?

      • You can check out how the number selection process works here: https://www.youtube.com/watch?v=Bup0TcbQeVs. As you can see, there’s really no way for any number to come up more often than any other number, unless one of the balls is somehow weighted differently. I wouldn’t be surprised if there are multiple sets of balls and the set that is actually used is somehow chosen at random behind the scenes.

        As for your question about numbers 16 and 33, you are falling into the “Hot Hand” fallacy, the thought that short term streaks will continue. There is no statistical basis for that theory.

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