We did the math to see if it's worth it to buy a ticket for the $415M Powerball jackpot

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Powerball Continues to Grow

The Powerball lottery drawing for Saturday evening has an estimated jackpot prize of $415 million.

While that's a huge amount of money, buying a ticket is still probably a losing proposition.

SEE ALSO: Powerball at $415M, 9th highest in US history

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

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 expected value is negative, then this game is a net loser for me.

RELATED: Powerball frenzy earlier this year

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January 2016 massive Powerball jackpot
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We did the math to see if it's worth it to buy a ticket for the $415M Powerball jackpot
7-Eleven store clerk M. Faroqui celebrates after learning the store sold a winning Powerball ticket on Wednesday, Jan. 13, 2016 in Chino Hills, Calif. One winning ticket was sold at the store located in suburban Los Angeles said Alex Traverso, a spokesman for California lottery. The identity of the winner is not yet known. (Will Lester/The Sun via AP)
7-Eleven store clerk M. Faroqui celebrates with customers after learning the store sold a winning Powerball ticket on Wednesday, Jan. 13, 2016 in Chino Hills, Calif. One winning ticket was sold at the store located in suburban Los Angeles said Alex Traverso, a spokesman for California lottery. The identity of the winner is not yet known. (Will Lester/The Sun via AP)
Hundreds gather outside the 7-Eleven, after it was announced the winning Powerball ticket was sold at the store, Wednesday, Jan. 13, 2016 in Chino Hills, Calif. One winning ticket was sold at the store located in suburban Los Angeles said Alex Traverso, a spokesman for California lottery. The identity of the winner is not yet known. (Will Lester/The Sun via AP)
People line up to buy Powerball lottery tickets at Kavanagh Liquors, Wednesday, Jan. 13, 2016, in San Lorenzo, Calif. The Powerball jackpot for Wednesday night's drawing is at least $1.5 billion, the largest lottery jackpot in the world. (AP Photo/Marcio Jose Sanchez)
Laura Woodward displays the Powerball lottery tickets she bought for her office pool at Lichine's Liquor store, Wednesday, Jan. 13, 2016, in Sacramento, Calif. The Powerball jackpot for Wednesday night's drawing to be over $1 billion, the largest lottery jackpot in the world.(AP Photo/Rich Pedroncelli)
People buy lottery tickets at the Primm Valley Casino Resorts Lotto Store just inside the California border Tuesday, Jan. 12, 2016, near Primm, Nev. The Powerball jackpot has grown to over $1 billion dollars for the next drawing on Wednesday. (AP Photo/John Locher)
Patrons line up to buy Powerball lottery tickets outside the Primm Valley Casino Resorts Lotto Store just inside the California border Tuesday, Jan. 12, 2016, near Primm, Nev. The Powerball jackpot has grown to over 1 billion dollars for the next drawing on Wednesday. (AP Photo/John Locher)
A sign shows the Powerball jackpot at a gas station Tuesday, Jan. 12, 2016, in Miami. The Powerball jackpot has grown to over $1 billion dollars for the next drawing on Wednesday. (AP Photo/Alan Diaz)
Snow falls on a billboard advertising the record $1.5 billion Powerball jackpot in downtown Pittsburgh, Jan. 12, 2016. (AP Photo/Gene J. Puskar)
This is a billboard advertising the record $1.5 billion Powerball jackpot in downtown Pittsburgh, Jan. 12, 2016. (AP Photo/Gene J. Puskar)
A tractor-trailer passes by a Kansas Lottery billboard in Topeka, Kan., Monday, Jan. 11, 2016. The Powerball jackpot has grown to over 1 billion dollars, and the drawing is still two days away. (AP Photo/Orlin Wagner)
A sign advertising the Powerball lottery hangs on a store in New York, Monday, Jan. 11, 2016. The jackpot is so big that billboards around the country have to advertise the prize as $999 million because they're not built to show billions. (AP Photo/Seth Wenig)
A Powerball ticket is sold in a truck stop in Carlisle, Pa., Monday, Jan. 11, 2016. The Powerball jackpot has grown to over 1 billion dollars. (AP Photo/Gene J. Puskar)
A customer shows his purchased Powerball tickets at a grocery store in Hialeah, Fla., Monday, Jan. 11, 2016. The Powerball jackpot has grown to over 1 billion dollars. (AP Photo/Alan Diaz)
A Pennsylvania lottery kiosk promotes the record $1.4 billion Powerball Jackpot in a mini-mart in Monroeville, Pa., Jan. 11, 2015. (AP Photo/Gene J. Puskar)
People walk past a sign advertising the Powerball lottery in New York, Monday, Jan. 11, 2016. The jackpot is so big that billboards around the country have to advertise the prize as $999 million because they're not built to show billions. (AP Photo/Seth Wenig)
A billboard outside Waverly, Neb., displays the top Powerball prize as $999 million, Monday, Jan. 11, 2016. The jackpot of over a billion dollars is so big that billboards around the country have to advertise the prize as $999 million because they're not built to show billions. (AP Photo/Nati Harnik)
FILE - In this Jan. 9, 2016 file photo, Powerball tickets are shown in San Lorenzo, Calif. No ticket matched all six Powerball numbers following the drawing for a record jackpot of nearly $950 million, lottery officials said early Sunday, Jan. 10, boosting the expected payout for the next drawing to a whopping $1.3 billion. (AP Photo/Marcio Jose Sanchez)
Kimeron Ruszovan as she displays her Powerball quick picks purchased Saturday, Jan. 9, 2016, in Sacramento, Calif. Powerball sales are at record numbers for Saturday nights drawing.(AP Photo/Rich Pedroncelli)
Sergio Ponce looks over his Powerball quick picks purchased Saturday, Jan. 9, 2016, in Sacramento, Calif. Powerball sales are at record numbers for Saturday nights drawing.(AP Photo/Rich Pedroncelli)
Francisco Figueroa, second from right, buys Powerball lottery tickets at the Blue Bird Liquor store in Hawthorne, Calif., on Friday, Jan. 8, 2016. Lottery officials confirmed Friday that Saturday nightâs drawing will be for a record $800 million. The ever-increasing jackpot is a result of strong national sales ever since the jackpot was near $400 million just days ago. (AP Photo/Nick Ut)
Jay Suthar works the lottery machine at Pine Liquors in Fort Washington, Friday, Jan. 8, 2016. With Powerball sales doubling previous records, the odds are growing that someone will win Saturdayâs $800 million jackpot, but if no one matches all the lottery numbers, next weekâs drawing is expected to soar past $1 billion. (AP Photo/Alex Brandon)
A Powerball lottery card rests on a table, Friday, Jan. 8, 2016, in Lyndhurst, Ohio. Lottery officials say sales have been robust since no one won Wednesday's estimated $500 million jackpot. It will be the biggest U.S. jackpot ever, beyond the previous record of a $656 million Mega Millions prize won in 2012 by players in Kansas, Illinois and Maryland. (AP Photo/Tony Dejak)
A woman purchases a Powerball lottery ticket at a convenience store in Washington, DC, January 7, 2016. Lottery officials predict Saturday's jackpot will reach $700 million, the largest in history. AFP PHOTO / SAUL LOEB / AFP / SAUL LOEB (Photo credit should read SAUL LOEB/AFP/Getty Images)
A sign in the window of a liquor store shows the Powerball lottery jackpot at $700 million in Washington, DC, on January 7, 2016. The largest jackpot in lottery history, a whopping $700 million, is up for grabs in the United States on Saturday, driving feverish excitement among lotto players dreaming of becoming millionaires. AFP PHOTO/NICHOLAS KAMM / AFP / NICHOLAS KAMM (Photo credit should read NICHOLAS KAMM/AFP/Getty Images)
A billboard displays the estimated jackpot for the Powerball lottery, Wednesday, Jan. 6, 2016, in Miami. The estimated Powerball jackpot for Wednesday night has soared to $500 million. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Alan Diaz)
Jose Garrido shows his Powerball tickets, Wednesday, Jan. 6, 2016, at a local grocery store in Hialeah, Fla. The estimated Powerball jackpot for Wednesday night has soared to $500 million. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Alan Diaz)
A clerk hands over a Powerball ticket to a customer, Wednesday, Jan. 6, 2016, at a local grocery store in Hialeah, Fla. The estimated Powerball jackpot for Wednesday night has soared to $500 million. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Alan Diaz)
CHICAGO, IL - FEBRUARY 11: A Powerball lottery ticket is printed for a customer at a 7-Eleven store on February 11, 2015 in Chicago, Illinois. Ticket sales have caused the jackpot to grow $500 million, one of the largest in the game's history. (Photo by Scott Olson/Getty Images)
CHICAGO, IL - FEBRUARY 11: Kirk Cook rings up a Powerball lottery ticket sale at a 7-Eleven store on February 11, 2015 in Chicago, Illinois. Ticket sales have caused the jackpot to grow $500 million, one of the largest in the game's history. (Photo by Scott Olson/Getty Images)
DUNKIRK, MD - JANUARY 06: With the Jackpot now at 500 million, Sherrie Haines sells a Powerball ticket to Robert Sweeney at the BP gas station, January 6, 2015 in Dunkirk, Maryland. People are visitingÃlottery counters across the area with hopes of hitting it big in tonights Powerball drawing. (Photo by Mark Wilson/Getty Images)
DUNKIRK, MD - JANUARY 06: With the Jackpot now at 500 million, Gale Call (L) and Sherrie Haines (C) sell a Powerball ticket to Mike Nastasi (R) at the BP gas station, January 6, 2015 in Dunkirk, Maryland. People are visitingÃlottery counters across the area with hopes of hitting it big in tonights Powerball drawing. (Photo by Mark Wilson/Getty Images)
A person purchase Powerball lottery tickets from a newsstand Wednesday, Jan. 6, 2016, in Philadelphia. Players will have a chance Wednesday night at the biggest lottery prize in nearly a year. (AP Photo/Matt Rourke)
A man purchases a Powerball lottery ticket at a liquor store in Washington, DC, January 4, 2016. Lottery officials predict the January 6 jackpot will reach $400 million, one of the largest in the game's history. AFP PHOTO / SAUL LOEB / AFP / SAUL LOEB (Photo credit should read SAUL LOEB/AFP/Getty Images)
FILE - In this Tuesday, Jan. 5, 2016 file photo, a sign shows the estimated Powerball jackpot in Spring, Texas. Players will have a chance Wednesday night at the biggest lottery prize in nearly a year. (AP Photo/David J. Phillip)
Powerball selection number selection slips are pictured at a store in Oklahoma City, Tuesday, Jan. 5, 2016. The estimated Powerball jackpot for Wednesday night has soared to $450 million. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Sue Ogrocki)
A billboards displays the current worth of a winning Powerball lottery ticket Tuesday, Jan. 5, 2016, in Springfield, Ill. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Seth Perlman)
A sign advertising Powerball tickets is pictured next to a display of instant lottery tickets in Oklahoma City, Tuesday, Jan. 5, 2016. The estimated Powerball jackpot for Wednesday night has soared to $450 million. (AP Photo/Sue Ogrocki)
Chris Huntsman, of Tulsa, Okla., displays his Powerball ticket at a store in Oklahoma City, Tuesday, Jan. 5, 2016. The estimated Powerball jackpot for Wednesday night has soared to $450 million. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Sue Ogrocki)
Jermaine Emory Mallory fills out a Powerball number selection slip at a store in Oklahoma City, Tuesday, Jan. 5, 2016. The estimated Powerball jackpot for Wednesday night has soared to $450 million. No one has won the big prize in two months and officials say the jackpot is now the sixth-largest ever in North America. Players will have a chance Wednesday night at the biggest lottery prize in nearly a year. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Sue Ogrocki)
A billboard displays the estimated jackpot for the Powerball lottery in Richmond, Va., Tuesday, Jan. 5, 2016. Players will have a chance Wednesday night at the biggest lottery prize in nearly a year. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Steve Helber)
A convenience store employee holds a Powerball lottery ticket in Richmond, Va., Tuesday, Jan. 5, 2016. Multi-State Lottery Association officials say the Powerball jackpot is up to $450 million for Wednesday night's drawing. It's the largest lottery jackpot in nearly a year and the sixth-largest ever in North America. (AP Photo/Steve Helber)
Powerball play slips are stacked up in a store in Richmond, Va., Tuesday, Jan. 5, 2016. Players will have a chance Wednesday night at the biggest lottery prize in nearly a year. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Steve Helber)
Jermaine Emory Mallory displays his Powerball ticket at a store in Oklahoma City, Tuesday, Jan. 5, 2016. The estimated Powerball jackpot for Wednesday night has soared to $450 million. Players will have a chance Wednesday night at the biggest lottery prize in nearly a year. The last time Powerball had grown this large was in February 2015, when three winners split a $564.1 million prize. (AP Photo/Sue Ogrocki)
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Powerball and similar lotteries are a wonderful example of this kind of random process. As of October 2015 in Powerball, five white balls are drawn from a drum with 69 balls, and one red ball is drawn from a drum with 26 balls. Prizes are then given out based on how many of a player's chosen numbers match the numbers written on the balls. Match all five white balls and the red Powerball, and you win the jackpot. In addition, there are several smaller prizes won for matching some subset of the drawn numbers.

Powerball's website helpfully provides a list of the odds and prizes for each of the possible outcomes. We can use those probabilities and prize sizes to evaluate the expected value of a $2 Powerball ticket. Take each prize, subtract the price of our ticket, multiply the net return by the probability of winning, and add all those values up to get our expected value:

main powerball odds

Thomson Reuters

Already, we are in trouble. Given the overwhelming likelihood — a 96% probability — that we wind up matching absolutely nothing and therefore wind up just throwing our $2 out the window, our expected value ends up being $0.26 in the red. We would need a prize of around $500 million to get a positive expected value.

Annuity vs. lump sum

Of course, even that scenario with a $500 million prize and a positive expected value is a vast oversimplification. First, the headline $415 million grand prize is paid out as an annuity: Rather than getting the whole amount all at once, you get the $415 million spread out in smaller — but still multimillion-dollar — annual payments over the course of 30 years. If you choose to take the entire cash prize at one time instead, you get much less money up front: The cash payout value at the time of writing is $269.7 million.

Looking at the lump sum, we are doing even worse than with the headline annuity, with an expected value of -$0.76:

lump sum odds

Thomson Reuters

The question of whether to take the annuity or the cash is somewhat nuanced. Powerball points out on their FAQ site that in the case of the annuity, the state lottery commission invests the cash sum tax-free, and you only pay taxes as you receive your annual payments, whereas with the cash payment, you have to pay the entirety of taxes all at once.

On the other hand, the state is investing the cash somewhat conservatively, in a mix of various US government and agency securities. It's quite possible, although risky, to get a larger return on the cash sum if it's invested wisely. Further, having more money today is frequently better than taking in money over a long period of time, 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

As mentioned above, there's the important 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 only taking home half of our potential prizes, we move even more solidly into negative expected-value territory, making our Powerball "investment" a very bad idea:

annuity after tax main

Thomson Reuters

The hit to halving the cash one-time prize is equally devastating:

lump sum after tax

Thomson Reuters

When you consider the expected value of a Powerball ticket, you can see that the lottery is a pretty bad "investment."

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