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The Powerball jackpot is $625 million. We did the math to see if you should buy a ticket

The Powerball jackpot after nobody won in Wednesday evening’s drawing is up to $625 million .

That’s a pretty big prize. However, taking a closer look at the underlying math of the lottery shows that it’s probably a bad idea to buy a ticket.

Consider the expected value

When trying to evaluate the outcome of a risky, probabilistic event like the lottery, one of the first things to look at is expected value.

Expected value is helpful for assessing gambling outcomes. If my expected value for playing the game, based on the cost of playing and the probabilities of winning different prizes, is positive, then in the long run, the game will make me money. If the expected value is negative, then this game is a net loser for me.

Expected values combine both probabilities and prize values to give us a better sense of the value of a game. As an example, consider a simple game where we flip a coin. If the coin is heads, you pay me $2. If it’s tails, I pay you $1.

Intuitively, this game seems like a bad idea for you, and expected value shows why.

To calculate the expected value, we multiply together the probability of each outcome by the value of each outcome and add them together. In our coin toss game, there’s a 1/2 chance you lose $2, and a 1/2 chance you win $1, so your expected value is (1/2 × -$2) + (1/2 × $1) which gives us -$1 + $0.50, for a final expected value of -$0.50 for you.

Lotteries are a great example of this kind of probabilistic process. In Powerball, for each $2 ticket you buy, you choose five numbers from 1 to 69 from a collection of white balls and one red “powerball” number from 1 to 26. Prizes are based on how many of the player’s chosen numbers match those drawn.

Match all six numbers, and you win the jackpot. After that, there are smaller prizes for matching some subset of the numbers.

The Powerball website helpfully provides a list of the odds and prizes for the game’s possible outcomes. We can use those probabilities and prize sizes to evaluate the expected value of a $2 ticket.

The expected value of a randomly decided process is found by taking all the possible outcomes of the process, multiplying each outcome by its probability, and adding all those numbers. This gives us a long-run average value for our random process.

Take each prize, subtract the price of our ticket, multiply the net return by the probability of winning, and add all those values to get our expected value.

We end up with an expected value of $0.46. Since that falls above our breakeven point of zero, it might look like a Powerball ticket could be a viable investment.

But, if we look more closely at other aspects of the lottery, things get much worse.

Annuity versus lump sum

Looking at just the headline prize is a vast oversimplification.

First, the $625 million jackpot is paid out as an annuity, meaning that rather than getting the whole amount all at once, it’s spread out in smaller — but still multimillion-dollar — annual payments over 30 years.

If you choose instead to take the entire cash prize at one time, you get much less money up front: The cash payout value at the time of writing is $380.6 million.

If we take the lump sum, we end up seeing that the expected value of a ticket drops all the way to -$0.38, well below our breakeven point:

The question of whether to take the annuity or the cash is somewhat nuanced. The Powerball website says the annuity option’s payments increase by 5% each year, presumably keeping up with or exceeding inflation.

On the other hand, the state is investing the cash somewhat conservatively, in a mix of US government and agency securities. It’s quite possible, though risky, to get a larger return on the cash sum if it’s invested wisely.

Further, having more money today is generally better than taking in money over a long period, since a larger investment today will accumulate compound interest more quickly than smaller investments made over time. This is referred to as the time value of money.

Taxes make things much worse

In addition to comparing the annuity with the lump sum, there’s also the big caveat of taxes. While state income taxes vary, it’s possible that combined state, federal, and — in some jurisdictions — local taxes could take as much as half of the money.

Factoring this in, if we’re taking home only half of our potential prizes, our expected-value calculations move ever further into negative territory, suggesting that our Powerball investment would be a bad idea.

Here’s what we get from taking the annuity, after factoring in our back-of-the-envelope estimated 50% in taxes. The expected value drops to -$0.61:

Losing as much as half of the prize to taxes is just as devastating to the lump sum, dropping the expected value to a whopping -$1.03:

Even if you win, you might split the prize

Another problem is the possibility of multiple jackpot winners.

Bigger pots, especially those that draw significant media coverage, tend to bring in more lottery-ticket customers. And more people buying tickets means a greater chance that two or more will choose the magic numbers, leading to the prize being split equally among all winners.

It should be clear that this would be devastating to the expected value of a ticket. Calculating expected values factoring in the possibility of multiple winners is tricky, since this depends on the number of tickets sold, which we won’t know until after the drawing.

However, we saw the effect of cutting the jackpot in half when considering the effect of taxes. Considering the possibility of needing to do that again, buying a ticket is almost certainly a losing proposition if there’s a good chance we’d need to split the pot.

One thing we can calculate fairly easily is the probability of multiple winners based on the number of tickets sold.

The number of jackpot winners in a lottery is a textbook example of a binomial distribution, a formula from basic probability theory. If we repeat some probabilistic process some number of times, and each repetition has some fixed probability of “success” as opposed to “failure,” the binomial distribution tells us how likely we are to have a particular number of successes.

In our case, the process is filling out a lottery ticket, the number of repetitions is the number of tickets sold, and the probability of success is the 1-in-292,201,338 chance of getting a jackpot-winning ticket.

Using the binomial distribution, we can find the probability of splitting the jackpot based on the number of tickets sold.

It’s worth noting that the binomial model for the number of winners has an extra assumption: that lottery players are choosing their numbers at random. Of course, not every player will do this, and it’s possible some numbers are chosen more frequently than others. If one of these more popular numbers turns up at the next drawing, the odds of splitting the jackpot will be slightly higher. Still, the above graph gives us at least a good idea of the chances of a split jackpot.

Most Powerball drawings don’t have much risk of multiple winners — the average drawing in 2019 so far sold about 22.6 million tickets, according to our analysis of records from LottoReport.com, leaving only about a 0.3% chance of a split pot. Wednesday evening’s drawing sold about 71 million tickets, which meant there was a 2.5% chance of a split pot.

But, bigger prizes could draw in more players — the record $1.6 billion Mega Millions jackpot in October 2018 had about 370 million tickets sold before its last drawing.

The risk of splitting prizes leads to a conundrum: Ever bigger jackpots, which should lead to a better expected value of a ticket, could have the unintended consequence of bringing in too many new players, increasing the odds of a split jackpot and damaging the value of a ticket.

To anyone still playing the lottery despite all this, good luck!

The low probability of winning and the risk of splitting the prize in a big, highly covered game mean you'd probably lose money.

Powerball executive

Meet the latest winners from your favorite lottery games. Remember, it only takes one ticket to win!

Lerynne West / $343.9 Million

CLIVE, Iowa (Nov. 5, 2018) — A central Iowa woman stepped forward Monday to claim a $343.9 million Powerball jackpot, the largest lottery prize won to date in Iowa.

Robert Bailey / $343.8 Million

NEW YORK (Nov. 14, 2018) – Sixty-seven-year-old Robert Bailey has come forward to claim the $343.8 million winning ticket from the October 27, 2018 Powerball drawing. Bailey, of New York City, now has the distinction of winning the largest jackpot in New York Lottery history. His winning ticket matched all six of the numbers drawn; numbers he said he has been playing for over twenty-five years.

Nandlall Mangal / $245.6 Million

(Sept. 27, 2018) – Forty-two-year-old Nandlall Mangal, from Staten Island, has come forward to claim the $245.6 million sole jackpot-winning ticket from the August 11, 2018 Powerball drawing.

Mangal, said he purchases his Lottery tickets when the jackpots reach $100 million. “I was grocery shopping and knew the Powerball jackpot was big,” he explained. “I decided that was a good time to buy my tickets.”

Steven Nickell / $150.4 Million

Salem, Ore. (July 13, 2018) – Steven Nickell of Salem still can’t believe the ticket he had in his wallet for two weeks was the $150.4 million Powerball jackpot winner.

Nickell purchased the ticket at the Circle K on Liberty Street in Salem, and said that he usually picks up tickets, then checks them when he goes to lunch. After he had a burger, he scanned his tickets and realized he needed to go to the Lottery headquarters – he thought he had won at least $600.

Tayeb Souami / $315.3 Million

TRENTON (June 8, 2018) – New Jersey Lottery Acting Executive Director John M. White announced Tayeb Souami of Little Ferry in Bergen County as the sole winner of the Saturday, May 19, 2018 Powerball Jackpot.

Mr. Souami had the only jackpot winning ticket for the May 19th Powerball drawing, making him New Jersey’s newest “MULTI” millionaire. He has elected to take the cash option with a payout of $183.2 million.

The Moose Family Trust / $55.9 Million

For the second time in less than five months Powerball jackpot winnings poured into Acadiana, after Lake Charles tax attorney Russell J. Stutes Jr. claimed the $55.9 million March 24 jackpot prize Tuesday on behalf of his client, The Moose Family Trust.

Emerald Legacy Trust / $456.7 Million

Middletown, PA (April 27, 2018) – A jackpot-winning Powerball® ticket from the March 17, 2018, drawing has been claimed, the Pennsylvania Lottery announced today. The jackpot had an advertised annuity value of $456.7 million.

The claimant, Emerald Legacy Trust, opted to receive a cash prize of $273,959,698, less 24 percent federal and 3.07 percent state tax withholding. The after-tax prize amount is $199,798,807.75.

Good Karma Family 2018 Nominee Trust / $559.7 Million

CONCORD, N.H. – The winner of a $559.7 million New Hampshire Powerball jackpot has claimed her prize and is donating a combined $250,000 to Girls Inc. of New Hampshire and three chapters of End 68 Hours of Hunger. The winner claimed the $559.7 million prize through the Good Karma Family 2018 Nominee Trust facilitated by the law firm of Shaheen & Gordon, P.A with William Shaheen serving as trustee.

292 Family Partnership / $191.1 million

Louisiana – Ending growing speculation about the whereabouts of Louisiana’s 16th Powerball jackpot-winning ticket, worth an estimated $191.1 million, Lafayette attorney Jean C. Breaux Jr. appeared at Lottery headquarters this morning [Dec. 12, 2017]with it and claimed the prize via a Power of Attorney on behalf of his client, a three-member family partnership.

Judy F. / $133.2 million

Denver – Judy F. from Clifton just proved that lottery players should never give up on their lucky numbers. She and her husband Mack just became the third Colorado Powerball players to win a jackpot.

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Powerball executive Meet the latest winners from your favorite lottery games. Remember, it only takes one ticket to win! Lerynne West / $343.9 Million CLIVE, Iowa (Nov. 5, 2018) — A central ]]>