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Should You Play Powerball? Science Solves The Mystery

Record sales drove up the largest jackpot in U.S. history to a whopping $1.5 billion as people . [+] dreaming of riches flocked across state lines and international borders to buy tickets. (Credit: Kena Betancur/AFP/Getty Images)

Playing the lottery is the ultimate low-risk, high-reward scenario. If you lose, you’re only out a few dollars: the cost of your bet. But if you win, even though the odds are stacked against you, the payoff is a lifetime of easy living. This evening, Wednesday, January 13th, will mark the richest lottery jackpot in history, with the current Powerball jackpot at a whopping $1.5 billion, a new record in lottery games worldwide.

Less than 24 hours from the record Powerball drawing. (Credit: Powerball.com)

In order to win, you need to match five normal lottery numbers — white balls numbered 1-through-69 — plus the Powerball: a red ball numbered 1-through-26. Each Powerball ticket costs $2, plus you have the option to pay an extra $1 to activate the power play , a multiplier that increases your payout for non-jackpot prizes.

Dozens of people lined up outside of Kavanagh Liquors in San Lorenzo, Calif., a store that has had . [+] several multi-million dollar winners. (Credit: Justin Sullivan/Getty Images)

But what does mathematics have to say about this? In particular:

    What are your odds of winning each individual combination?

The idea of “worth it” is a subjective one to most people, but from a scientific/mathematical standpoint, it has a very particular meaning. It means that the amount you can expect to win, on average, is greater than the amount you have to bet in order to play. If a lottery ticket costs $1, for example, it would be worth it if:

  • You had a 51% chance of winning $2.
  • Or, you had a 0.1% chance of winning $1001.
  • Or, you had a 1-in-499,999 chance of winning $500,000.

While it wouldn’t be worth it if:

  • You had a 49% chance of winning $2.
  • Or, you had a 0.1% chance of winning $999.
  • Or, you had a 1-in-500,001 chance of winning $500,000.

Notice how small these differences are, but how in the earlier cases, you can expect to win more than you bet, while in the latter cases, you expect to bet more than you win, at least on average. Mathematicians call this ratio of how-much-you-win vs. how-much-you-bet the expected value of a problem. If your expected value is greater than 1.0, it’s worth it to play.

The less likely outcomes need to have a greater payoff to be worth it; a relatively likely outcome . [+] would need only a modest payout to be worth it. Image credit: Wikimedia Commons user Dan Kernler.

So what, then, does this mean for Powerball? In every game of Powerball, you get one ticket with five white numbers (out of 69 possible choices) and one red number (the Powerball, out of 26).

Let’s hit the first point we asked about: what are your odds of winning, with each individual combination highlighted? Here’s an infographic I made that breaks it down.

(Credit: E. Siegel, 2016)

Your odds of actually winning the Powerball jackpot are pretty slim: one in 292,201,338 . In fact, your odds of winning anything aren’t very good either, since the three most common results:

  • no matches of any type (65.23%),
  • one white ball and no powerball (27.18%), and
  • two white balls and no powerball (3.565%),

all pay out absolutely nothing, and add up to 95.98% of the possible results.

But that means, 4.02% of the time, you will win something. And if — on average — it pays out enough, it will be worth it to bet.

Image credit: screenshot from the official Powerball site, at . [+] http://www.powerball.com/powerball/pb_prizes.asp.

So if these are the possibilities for what the payout is, what does this mean in terms of expected value? In other words, each time you play the lottery, you have a small chance of winning a certain prize, and if you multiply your expected winnings by your odds of each option, you get your expected value for each option. (And remember, the cost of a ticket for the Powerball is $2.00.)

Let’s work it out! We’ll come back to the “Jackpot” in just a bit, because it’s a bit more complicated. But let’s take a look at all the other — more likely, but non-Jackpot — options first.

(Credit: E. Siegel, 2016)

For every $2 ticket you buy, you can expect to recoup, on average:

  • about $0.15 from the periodic $4 payouts,
  • about $0.02 from the periodic $7 payouts,
  • about $0.01 from the periodic $100 payouts,
  • about $0.05 from the periodic $50,000 payouts, and
  • about $0.09 from the periodic $1,000,000 payouts.

All told, the non-jackpot options make each ticket worth about $0.32, which is a far cry from the $2 you invested. This teaches us two things:

  1. It gives us the information we need to figure out how much the “Power Play” option is actually worth.
  2. It let’s us know how much the Jackpot needs to pay out in order for buying a Powerball ticket to be “worth it,” mathematically.

First, the Power Play.

Screenshot and headers from the official Powerball site. (Credit: Powerball.com)

The Power Play option — which costs an extra $1.00, turning a $2 ticket into a $3 ticket — does the following:

  • has no effect on the Jackpot/Grand Prize,
  • always doubles the payout of the second-most-lucrative prize, and
  • has a 1-in-1.75 chance of doubling the other prizes, a 1-in-3.23 chance of tripling them, and a 1-in-14 chance of quadrupling or a 1-in-21 chance of quintupling them.
  • or, if the 10x multiplier is active, reducing the chances of all the other options a tiny bit, and adding in a 1-in-43 chance of multiplying the other prizes by ten.

So what’s the extra expected payoff for this extra $1 investment?

It takes the non-jackpot options, on average, from being worth $0.32 up to being worth $0.81. This means you’re spending an extra $1.00 to increase your expected payout by $0.49, a lousy deal any way you slice it. In fact, even if you happened to hit the 5x option, which happens only about 5% of the time, you only up your expected winnings to $1.34 for the non-jackpot options, which increases your winnings by a mere $1.02. That’s what you need to make it “worth” grabbing the Power Play option: a guaranteed 5x multiplier or more. The fact that the second-biggest-payout is only doubled, no matter what the Power Play multiplier happens to be, makes this a raw deal any way you slice it.

In other words, you should never take the Power Play option.

So finally, we come to the big prize: the Jackpot, or the Grand Prize, which you win by hitting all five numbers plus the Powerball, something that has a one-in-292,201,338 chance of happening.

Frank and Shirley Capaci of Streamwood, Il won $104 Million in the Powerball lottery in May. . [+] (Credit: Daniel Lippitt/AFP/Getty Images)

You would think, perhaps, that if a Powerball ticket costs you $2, and you have an expected value of $0.32 from the rest of the ticket, as long as your expected value is $1.68 or higher from the Powerball Grand Prize, you’ll come out ahead, and you should play.

That logic is sound, by the way: you’re right on! If your ticket is “worth” more than $2 total, of course you should spend $2 on it.

But you might then take the next step, and say, “since my odds of winning the Jackpot are 1-in- 292,201,338 , all I need to do is find what payout corresponds to that expected value and, if the Jackpot is more than that, I’ll play.” Finding that value isn’t so hard: it’s $245,449,123.92. But if you play the lottery when the Jackpot is that size, you’ll come out behind, still, for two reasons.

Even after the taxman calls, the Powerball jackpot could make the lucky winner wealthier than the . [+] likes of British boy band One Direction, soccer star Lionel Messi, Swiss tennis champion Roger Federer and U.S. superstar Beyonce. (Credit: Kena Betancur/AFP/Getty Images)

One of those reasons is taxes. That’s right: you don’t get to keep 100% of your winnings, although you are responsible for 100% of the costs of the tickets. The “advertised jackpot” is how much you’d get, pre-taxes, if you deferred the payments out over a long period of time.

If, instead, you took a lump sum payment (which is drastically reduced to about 63% of the advertised value), and then paid (state and federal) taxes on that, you’d discover something shocking: you only get to keep 37.2% of the Grand Prize’s value! (Dependent on your state’s taxes, which are estimated to be around 6%.) In fact, state and federal taxes would be a big deal for the other, smaller prizes we talked about earlier, which would bump the $1,000,000 prize down to be worth only about $590,000, and would reduce the value of the “other” payouts from being worth a total of $0.36 to about $0.26, a much worse deal.

In order to hit the “break-even” point, and have a $2 ticket actually be worth $2 when you include taxes, you’d need to have the Jackpot come in at a value of a whopping $1.4 billion, which only today’s Powerball Jackpot has crested in all of history.

But even that makes an assumption: that if you win, that your winning ticket will be the only winning ticket, which runs counter to the facts. As you well know, people are far more likely to buy lottery tickets when the Jackpot is huge. While you might not think this would mean so much with incredibly long odds, once you start selling more than about 70 million tickets, the odds of more than one person hitting the jackpot rise extremely quickly, as Jeremy Elson’s research into the matter has demonstrated.

(Credit: Jeremy Elson, via http://www.circlemud.org/jelson/megamillions/)

Based on various Jackpots that we’ve seen, once the advertised Jackpot passes about $300 million, ticket sales rise exponentially, with the largest jackpots resulting in hundreds of millions of tickets sold.

The thing is, not everyone gets a unique set of numbers: many tickets share the same numbers, so that if you get to, say, an advertised Jackpot of around $700 million, you’d expect to sell about 190 million tickets for the drawing. Even though the odds of any individual ticket hitting the Jackpot is 1-in- 292,201,338 , the odds that only one person will win that Jackpot is much lower than you’d think: about 37%. There’d be a 34% chance that no one would hit the Jackpot, and a 29% chance that two or more people would win.

What’s crazy — and unintuitive — is that as the Jackpot rises higher and higher, because more and more tickets get sold, the less valuable each ticket becomes! A ticket sold for a $1,500 million (or $1.5 billion) Jackpot, in fact, would only be worth about half as much as a ticket sold for a $500 million Jackpot, because you’d most likely have to split the Jackpot, even if you won, with between three and seven other people. And that’s probably what’s going to happen tonight.

These figures are for the “old” Powerball lottery, superseded about 12 months ago to increase . [+] jackpots. (Credit: Jeremy Elson, via http://www.circlemud.org/jelson/megamillions)

When you take both taxes and split Jackpots into account, you find that even at its maximum value, a $2 Powerball ticket is really only worth about $0.852, or just 43% of what you paid for it.

It’s long been said that lottery tickets are a tax on those who can’t do math, and now you’ve seen the mathematical proof of that. If you want to buy a Powerball ticket for fun, by all means, go right ahead. Just be aware that for every $2 you spend, you’re donating about $1.15 to whatever government programs that Powerball supports, and only betting $0.85 on whatever you might win in an otherwise “fair” lottery.

As the jackpot crests $1 billion for the first time, it’s time to examine the scientific answer to the ultimate question: is it worth it to play?

Frequently Asked Questions

1. What is POWERBALL ® ? POWERBALL is an exciting, multi-state, multimillion-dollar-jackpot game. It is a product of the Multi-State Lottery Association (MUSL) and is offered in 48 lottery jurisdictions (including Florida). All POWERBALL member lotteries contribute a percentage of sales to a shared top-prize pool so jackpots can grow to hundreds of millions of dollars. The game’s starting jackpot is $20 million and is the largest in the world. POWERBALL has nine prize levels with lower-tier prizes ranging from $4 to $1 million – and up to $2 million with Power Play ® . 2. What other lotteries offer POWERBALL? POWERBALL is played in 45 states, the District of Columbia, Puerto Rico and the U.S. Virgin Islands. For a complete list of states that offer POWERBALL visit www.powerball.com. 3. Will the revenue generated from Florida ticket sales stay in the state? As with all other games offered by the Florida Lottery, all profits from POWERBALL ticket sales in Florida will be transferred to the Educational Enhancement Trust Fund to benefit education in the state. 4. When did POWERBALL sales begin in Florida? POWERBALL ticket sales in Florida began on January 4, 2009. 5. When did the first POWERBALL drawing in Florida take place? When the Florida Lottery decided to join POWERBALL, the decision was also made to move the drawings to Florida. Therefore, the first POWERBALL drawing that included Florida players, was also the first POWERBALL drawing conducted in Florida, and it occurred on Wednesday, January 7, 2009. 6. How much do POWERBALL tickets cost? POWERBALL tickets cost $2 per play. POWERBALL with Power Play tickets cost $3 per play. 7. What is the largest POWERBALL jackpot to date? The largest POWERBALL jackpot to date was a whopping $1.5 billion shared by winners in Florida, Tennessee and California on January 13, 2016. 8. Who sets the POWERBALL jackpot and how is it calculated? The advertised POWERBALL jackpot is set by MUSL for each drawing. Jackpots are calculated based on anticipated sales for that drawing, funds rolled over from previous drawings and current interest rates for long-term investments. For every $2 ticket sold, 34.0066% is allocated to the jackpot pool. 9. What is Power Play? POWERBALL players can add Power Play for $1 more per play to multiply their winnings. With Power Play, players can win $2 million for matching the five white balls or multiply their other non-jackpot prizes by 2X, 3X, 4X, or 5X. When the advertised jackpot is $150 million or less a 10X multiplier will also be added. A player must choose the Power Play option on his/her POWERBALL playslip or tell the retailer that he/she wishes to purchase a Power Play ticket. The multiplier number will be selected just before the draw. 10. Where can I view the POWERBALL drawings? You may view the POWERBALL drawings on official Florida Lottery TV carrier stations in major markets throughout Florida. Carrier stations will air the drawings, either live or within minutes of the live drawings, or display on-screen the winning numbers following each drawing. Viewing times will vary by station. POWERBALL drawings may also be viewed via Webcasts on this website.

POWERBALL winning numbers will be available on this website, at Florida Lottery retailers statewide, and by phone at (850) 921-PLAY. Also, you may register on this website to receive POWERBALL winning numbers and other Lottery information by email or text message. 11. How do you play POWERBALL? At any Florida Lottery retail location, you can either mark your selections on a POWERBALL playslip or tell the retailer the numbers you wish to play. On the playslip, select five numbers from 1 to 69 in the upper play area and one Powerball number from 1 to 26 in the lower play area. You may Quick Pick some or all of your numbers by marking the QP box in the upper and/or the lower play areas of the playslip. Make your selections carefully, because POWERBALL TICKETS CANNOT BE CANCELED. 12. How do you play Power Play? Simply mark the Power Play box located on the left side of the POWERBALL playslip. Power Play will apply to all panels and Quick Pick boxes you have marked on the playslip. Power Play will add $1 more per play, so check your selections carefully before giving the playslip to the retailer. If you tell the retailer your POWERBALL selections, make sure to ask for Power Play. 13. How will I know which plays on my POWERBALL ticket include Power Play? Power Play will apply to all plays on a POWERBALL ticket. POWERBALL with Power Play tickets will display “POWER PLAY YES” and POWERBALL tickets with no Power Play will display “POWER PLAY NO.” 14. Can I advance play POWERBALL? If so, how many draws? Yes, you may advance play POWERBALL for up to 52 consecutive draws. If you select Power Play, it will apply to all drawings. 15. How late can I purchase a POWERBALL ticket on a drawing night? On drawing nights, you will be able to purchase POWERBALL tickets until game sales close at 10:00 p.m. ET. MUSL requires adequate time to ensure that the draw data has been securely received from all 47 member lotteries before the POWERBALL drawing is held at 10:59 p.m. ET.

There will be a 5-minute sales break between 10:00 p.m. and 10:05 p.m. ET on the night of the POWERBALL drawings, and sales after 10:05 p.m. ET will be for the next drawing. 16. Will my POWERBALL playslip from another state work in Florida? No, all POWERBALL states use different playslips that work exclusively on their gaming systems. 17. On the POWERBALL ticket, how is it possible to have the same number twice on one line? Since you select five numbers from 1 through 69 and one Powerball number from 1 through 26, it is possible to repeat a number for your Powerball number. The letters “PB” will indicate the Powerball number on your ticket. POWERBALL numbers are drawn from two separate sets of balls in which five white balls and one red Powerball are selected at random. 18. Can I purchase Florida POWERBALL tickets through the Internet or by mail if I live out of the state/country? No. Florida POWERBALL tickets may only be purchased from the Florida Lottery through terminals at authorized Florida Lottery retailers. Although POWERBALL is sold nationally, each state can only sell POWERBALL within their state. 19. How do you win POWERBALL? In the official POWERBALL drawing conducted by MUSL, five white balls (numbered 1 through 69) and one Powerball (numbered 1 through 26) are randomly selected. Players win the jackpot by matching, in any order, the five numbers drawn from the first set of balls and the Powerball drawn from the second set. Players win other great POWERBALL prizes, ranging from $4 up to $1 million, by matching non-jackpot winning combinations.

Prior to the POWERBALL drawing, a Power Play multiplier number is selected and displayed during the draw. Players who added Power Play to their purchase, will increase their winnings on the third through ninth prize levels by the multiplier number selected for that draw. 20. What are the odds of winning a POWERBALL prize or the POWERBALL jackpot? The overall odds of winning a prize are 1 in 24.9 and the odds of winning the jackpot are 1 in 292,201,338. 21. If the red Powerball number is drawn from a set of 26, why are the odds of matching only the Powerball number not 1 in 26? POWERBALL numbers are drawn from two sets of numbers, so the odds of winning are calculated by combining the odds for both sets of numbers for all prize levels. The odds for the ninth prize level (for matching just the Powerball) are calculated by combining the odds of selecting the Powerball and the odds of not selecting any of the five numbers from the first set of numbers drawn. 22. Are POWERBALL prizes shared? POWERBALL jackpots are shared among all jackpot winners in all POWERBALL states. All other prize levels are set prizes. 23. How much can I win with Power Play?

The match 5 prize is $2 million. All other non-jackpot prizes are increased by the multiplier number – 2, 3, 4, or 5 – selected just before each draw. The multiplier number 10 is included when the advertised jackpot is $150 million or less. 24. Is the $2 million in the match 5 Power Play prize paid in cash or as an annuity? The $2 million Power Play prize will be a one-time cash payment less applicable taxes. 25. Do I have to claim my Florida POWERBALL jackpot prize at MUSL headquarters in Des Moines, Iowa? No, all Florida POWERBALL prizes including the jackpot are paid in Florida, like all other games offered by the Florida Lottery. 26. Can I claim a Florida POWERBALL winning ticket in another POWERBALL state? No. All prizes must be claimed in the state where the winning ticket was sold. 27. When do Florida POWERBALL tickets expire? Florida POWERBALL tickets expire 180 days from the winning drawing date, just like all other terminal games offered by the Florida Lottery. 28. Does the POWERBALL jackpot have a cash option? Yes. Jackpot winners may choose to receive their portion of a jackpot prize as an annuity payment or a one-time, lump-sum payment (or Cash Option). Jackpot prizes must be claimed within the first 60 days after the winning drawing date to receive the Cash Option. 29. How long will it take to get my POWERBALL jackpot prize payment? MUSL requires two weeks to receive wire transfers of funds from all 47 POWERBALL members, so the earliest you may receive your POWERBALL jackpot prize is on the 15th day after the winning drawing date. 30. What happens to the money if a POWERBALL jackpot ticket is won in Florida but is not claimed? At the end of the winning state’s claim period (180 days in Florida), the funds to pay a jackpot that goes unclaimed will be returned to the lottery members in their proportion of sales for the jackpot rollover series.

In Florida, 80 percent of all unclaimed jackpot prizes are transferred to the Educational Enhancement Trust Fund to benefit education in the state. 31. Who gets the unclaimed prize money for non-jackpot POWERBALL prizes? All funds for unclaimed POWERBALL non-jackpot prizes stay in the state where the winning tickets were sold. 32. What happens if a POWERBALL jackpot winner dies before all payments have been made? The POWERBALL annuity prize will go into the deceased winner’s estate. Annual prize payments can continue to be paid to heirs, or the annuity may be cashed out by the winner’s estate.

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Frequently Asked Questions 1. What is POWERBALL ® ? POWERBALL is an exciting, multi-state, multimillion-dollar-jackpot game. It is a product of the Multi-State Lottery Association (MUSL) and is ]]>