# Should You Play Powerball or Mega Millions? Probability & Expected Return

TR Smith is a product designer and former teacher who uses math in her work every day.

This article has been recently edited to reflect the most recent changes in both lotteries' structures.

Powerball and Mega Millions are the two largest lotteries in the US in terms of the number of players, revenue generated, and jackpot size. (See link for more detailed probability calculations.) They both offer astronomical odds against winning, but when jackpots reach the high tens of millions and hundreds of millions, lottery fever sets in and few can resist throwing away \$1 or \$2 for a chance to daydream about what they could do with all those millions. Most states offer both lotteries, so it is natural to wonder which lottery is "better."

The answer depends on how you define "better." You can compare the two lotteries by examining the overall probability of winning any prize, the expected return for every \$1 wagered, the typical sizes of the prizes, or even the cost of a ticket. Analyzing each of these factors will give you a clearer picture of which lottery best suits your interests as a regular lottery player or casual lottery ticket buyer.

## Powerball Odds and Prizes

A Powerball lottery ticket costs \$2 for a single play, and \$3 for a single play if you choose the Power Play option. With Power Play, if you make a partial match and win a lower-tier prize, your prize is multiplied by a certain amount called the multiplier.

The multiplier value is randomly determined at each drawing and may increase prizes by a factor of 2, 3, 4, 5, or 10 when the jackpot is under \$150 million. When the jackpot is at least \$150 million, the 10 multiplier is not put in play and only 2, 3, 4, or 5 will be drawn. The multipliers are not uniformly distributed; 2 is much more likely to be drawn than 3, which is much more likely than 4, etc. The average expected multiplier value works out to be either 2.59524 when the "10" is not in play, or 2.76744 when the "10" is in play.

Regardless of the multiplier value drawn, the second prize will only increase to \$2 million with the Power Play option. With the most recent change in Powerball's structure, you can expect that most drawings will have a jackpot of at least \$150 million, so the 10-valued multiplier will not be in play very often. Here are the odds and prizes for each Powerball prize level.

Match
Odds
Regular Prize
5/5 + PB
1 in 292,201,338
share of jackpot
5/5
1 in 11,688,053.25
\$1,000,000
4/5 + PB
1 in 913,129.18125
\$50,000
4/5
1 in 36,525.16725
\$100
3/5 + PB
1 in 14,494.11399
\$100
3/5
1 in 579.76456
\$7
2/5 + PB
1 in 701.3281
\$7
1/5 + PB
1 in 91.97746
\$4
only PB
1 in 38.32394
\$4
Overall Odds
1 in 24.87

## Mega Millions Odds and Prizes

The Mega Millions lottery has a similar odds and payout structure. Tickets only cost \$1 for a single play. For \$2 you get a one play plus the Megaplier option. If you purchase the Megaplier option with your lottery ticket, your non-jackpot prizes are multiplied by 2, 3, 4, or 5.

The multiplier value (called the megaplier) is selected randomly during each drawing, but the four values do not have an equal chance of occurring. A megaplier value of 5 is more likely to to occur than a 3, which is more likely than 4, which is more likely than a 2. The average expected multiplier works out to be 3.86667 (the weighted mean of 2, 3, 4, and 5). Here are the odds and prizes for Mega Millions.

Match
Odds
Regular Prize
w/ Megaplier
5/5 + MB
1 in 258,890,850
share of jackpot
share of jackpot
5/5
1 in 18,492,203.57143
\$1,000,000
\$2,000,000 or \$3,000,000 or \$4,000,000 or \$5,000,000
4/5 + MB
1 in 739,688.14286
\$5,000
\$10,000 or \$15,000 or \$20,000 or \$25,000
4/5
1 in 52,834.86735
\$500
\$1,000 or \$1,500 or \$2,000 or \$2,500
3/5 + MB
1 in 10,720.11801
\$50
\$100 or \$150 or \$200 or \$250
3/5
1 in 765.72272
\$5
\$10 or \$15 or \$20 or \$25
2/5 + MB
1 in 472.94638
\$5
\$10 or \$15 or \$20 or \$25
1/5 + MB
1 in 56.47121
\$2
\$4 or \$6 or \$8 or \$10
only MB
1 in 21.39061
\$1
\$2 or \$3 or \$4 or \$5
Overall Odds
1 in 14.708

## Comparison of Powerball and Mega Millions

With the odds and prizes we can now compare various aspects of the two games with and without the multiplier options. The aspects we will look at are

• Odds of winning the jackpot
• Overall odds of winning any prize (including break-even prize)
• Expected return

Odds of winning jackpot: A regular \$1 Mega Millions ticket offers better odds of winning the grand lottery prize than a regular \$2 Powerball ticket. Paying the for multiplier option with either game (\$3 Powerball and \$2 Mega Millions) does not increase your chances.

Conclusion: If you want to optimize your jackpot odds per dollar, buy a regular Mega Millions tickets at \$1 rather than a \$2 Powerball ticket.

## Which lottery do you usually play?

See results

Overall odds of not losing money: The overall odds for winning any prize, including a break-even prize equal to the cost of your ticket, is 1 in 24.87 for the Powerball, and 1 in 14.708 for the Mega Millions. In other words, you have a 4.02% chance of not losing money with Powerball, but a 6.8% chance of not losing money with Mega Millions.

Conclusion: A Mega Millions tickets gives you better chance of not losing money, and it's half the cost of a Powerball ticket.

## Expected Return per Dollar

Analyzing the expected return per dollar is the way a serious or professional gambler analyzes a game of chance. To calculate the expected return on a lottery, you multiply each prize level by its probability (same as dividing by the odds), sum those products, then divide by the amount of the wager i.e. the ticket price. This gives you the expected return per dollar. In other words, it's how much you can expect to get paid for every dollar you spend on lottery tickets. (Subtracting the price of of the ticket gives you the expected net profit, which is another way to judge how bad a wager is.)

If you do this for Powerball, Powerball with Power Play, Mega Millions, and Mega Millions with Megaplier you get the following unadjusted expected returns:

Game
Unadjusted Expected Return per \$1 Wagered
Powerball
0.1599 + P/584402676 dollars
Powerball w/ Power Play
0.2597 + P/876604014 dollars
Mega Millions
0.1742 + M/258890850 dollars
Mega Millions w/ Megaplier
0.3368 + M/517781700 dollars

In the table, P stands for your share of the Powerball jackpot and M stands for your share of the Mega Millions jackpot. Remember, jackpots can be split among two or more winning tickets.

The Megaplier option expected return is calculated using an average multiplier value of 3.86667, while the Power Play option expected return is calculated using an average multiplier value of 2.59524. P and M are divided by very large numbers that come from the odds and the price of a ticket. Since P and M vary and are unpredictable, one cannot know the precise unadjusted value of the expected return. But we can compare the adjusted expected returns of all four lottery options.

To adjust these returns, we ignore the outlier event of winning the jackpot. Statistically this is extraordinarily unlikely event that can safely be disregarded. This gives us the adjusted expected returns

• Powerball: \$0.1599 per \$1
• Powerball w/ Power Play: \$0.2597 per \$1
• Mega Millions: \$0.1742 per \$1
• Mega Millions w/ Megaplier: \$0.3368 per \$1

As you can see, the expected returns are all under \$1, meaning you get back less than you "invest." Equivalently, the expected profits are negative, i.e., a loss. The regular Powerball ticket without the Power Play option has the lowest adjusted expected return on the dollar, while Mega Millions with Megaplier option has the highest adjusted expected return per dollar.

We can adjust these expected returns even further to exclude prize levels with odds that are worse than 1 in a million. This excludes the second-tier prizes from the calculations, since you can consider the second-tier prizes to be somewhat outlier events. This gives a new set of adjusted expected returns:

• Powerball: \$0.1172 per \$1
• Powerball w/ Power Play: \$0.2027 per \$1
• Mega Millions: \$0.1202 per \$1
• Mega Millions w/ Megaplier: \$0.2323 per \$1

With these calculations, the regular Powerball lottery has the worst expected return, while the Mega Millions with Megaplier has the best expected return.

Conclusion: To maximize the expected return per dollar, buy the \$2 Mega Millions ticket with the Megaplier option. With both methods of adjustment it is the better bet, or more accurately, the least bad bet.

Comparison to Hot Lotto: The smaller multi-state lottery game Hot Lotto has an adjusted expected return of \$0.2656 per \$1 bet, so it's a better bet than either Powerball or Mega Millions, with or without the Megaplier and Power Play options. Most state lotteries offer a higher return than Powerball and Mega Millions. However, the jackpots are smaller and less seductive.

## Use Lottery Simulations to Gain Intuition About Lottery Odds

To get a better sense of just how bad a bet either lottery is, you can run thousands of simulated random lottery draws with a random number generator or random lottery simulator. I ran a lottery number simulator for 74,892 drawings and obtained the following results for Powerball and Mega Millions:

Game
\$ Amount Spent on 74,892 Games
\$ Amount Won
Profit or Loss
\$ Return per \$1 Spent
Powerball
\$149,784
\$12,859
-\$136,925
\$0.0856
Powerball w/ Power Play
\$224,676
\$41,799
-\$182,877
\$0.1860
Mega Millions
\$74,892
\$6,074
-\$68,818
\$0.0811
Mega Millions w/ Megaplier
\$149,784
\$24,939
-\$124,845
\$0.1665

The returns generated by the simulator are consistent with the theoretical values computed above. Never in 74,892 simulations did I ever win anything better than a fourth prize level match. Considering that you have to pay taxes on your lottery earnings in the US, the true returns are even lower.

In comparison, keno lottery games have a return of at least \$0.60, and many are as high as \$0.70. Of course, they don't have the enormous jackpots that make Powerball and Mega Millions so enticing.

## Are "Lottery Strategy" Guides Any Better Than Picking Numbers Randomly?

Hundreds of self-styled lotto gurus claim that certain number combinations win more often than others, and they will tell you what those combinations are if you buy one of their systems for the low, low price of \$19.99. In truth, there are no systems that will ensure that you win more than you spend on lotteries. The best you can do is play lotteries with higher prize odds to lessen your losses. The vast majority of people who play the lottery are going to lose much more money than they ever win.

Lottery strategy guides are no more accurate at predicting winning numbers than a novelty pen that generates random lottery numbers, or a digital bio-feedback device that generates winning numbers based on your "body energy." Both items are pictured below.

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• Rafael 3 years ago

so basically, it's a sucker's bet either way.

• MomsTreasureChest 3 years ago

Thanks for all the great information! It's discouraging but I'll still be playing the mega millions and powerball lottery games, I figure someone is going to win, and ya gotta play to win...so wish me luck LoL!

• Teli 2 years ago

Thanks, this is very useful and well-explained.

• Will 2 years ago

Hello C-G,

I am working through a statistics and probability book and am trying to learn more about lottery probabilities. When I try to calculate the odds of winning powerball on my own, I get 1 / (59*58*57*56*55*35) = 1 / 21026821200, which is much larger than the odds you listed in your table. Similarly when I calculate the odds of MegaMillions. I'm trying to see where my logic is off. I figured you have 59 possible choices for the first ball, 58 for the second after the first is already selected, 57 for the third after the first two are taken out, 56 for the fourth, 55 for the fifth, and then 35 for the powerball. Multiplying 59*58*57*56*55*35 gives you 21026821200. Can you show where the misstep is?

Also I left a question comment on your Keno Probability article https://owlcation.com/stem/Keno-Lottery-Strategy...

• Author

TR Smith 2 years ago from Eastern Europe

Hi Will,

Thanks for your two comments. The detailed explanation of how to calculate both probabilities is given in the link in the first paragraph of this article. I did not want to clutter this article with repeat information. Essentially, you need to divide your number by 5*4*3*2*1 = 120 to correct the overcount. This is because the product 59*58*57*56*55*35 gives the number of ordered number sets. But in a lottery, the order in which the balls are drawn (the first 5 balls) does not matter.

• Will 2 years ago

Thanks for the explanation. I have another question if you don't mind, how do you make a lottery with a certain desired probability of winning. Say, a lottery like the Powerball where the probability of winning the jackpot is exactly 1/12,000,000?

• Author

TR Smith 2 years ago from Eastern Europe

If the game is structured as drawing five balls numbered 1 to X and one ball numbered 1 to Y, then you need to find solutions to the equation

X(X-1)(X-2)(X-3)(X-4)Y/120 = 12,000,000

If it's drawing four balls instead of five, the equation to solve is

X(X-1)(X-2)(X-3)Y/24 = 12,000,000

These equations might not have solutions where both X and Y are integers, but sometimes you can get close to your target. For example, in a lottery where you choose five regular from 1 to 41 and one powerball from 1 to 16, the probability of winning is

1/(41*40*39*38*37*16/120) = 1/11990368

which is pretty close to the target of 1/12000000.

Also, a lottery with five balls selected from 1 to 35 and one powerball selected from 1 to 37 has a probability of

1/(35*34*33*32*31*37/120) = 1/12011384

• Jean 2 years ago

Where I live you can play both. I always had a gut feeling Mega Millions was the better bet for some reason (ok, the less bad bet, lol). I'm glad to see it confirmed mathematically.

• Jonathan Wirthlin 2 years ago

I am further interested in one more statistic. Increased ticket sales at the highest jackpots also mean greater chances of sharing the pot. This means the possibility of jackpot return is not linearly upwards with the jackpot growth. Alternately, one ticket at the lowest pot has a lower expected return. I'm interested in the peak jackpot to play to maximize jackpot possible winnings. (The high jackpot that has low chances of sharing with other frenzied ticket purchasers.) Please provide.

• Author

TR Smith 2 years ago from Eastern Europe

Hi JW,

Thanks for the interesting question. I will address this in another lottery math article.

• Caral 2 years ago

Suppose there's a lottery that's a simple pick 6 out of 50 type of game, no extra balls. There are prizes for matching 6, 5, 4 or 3 of the drawn numbers. What do the prize amounts have to be so that the expected return is \$0.40? Assume ticket costs a buck. Thanks if you can help me out with this math problem!

• zan 22 months ago

Powerball is up to 110 million at this moment but it says cash value is 70.5 million. What are these two different numbers? Is the prize 110M or 70.5M? Thanks.

• Author

TR Smith 22 months ago from Eastern Europe

Caral: Thanks for the interesting question. There are many possible solutions which you can find just with trial and error. One prize structure that fits the bill is to award \$2000, \$15000, \$100000, and \$10000000 for matching 3, 4, 5 or 6 numbers respectively. If the tickets are \$1, this yields an expected return of \$0.4085.

Zan: The cash value is the lower number and the annuity value is the higher number. The annuity option doles out the jackpot prize money in increments every month or every year.

• zan 22 months ago

After several drawings with no winner the jackpot is now up to 138 million (annuity) and 86.3 million (cash value). When it was 110 and 70.5, the ratio of annuity to cash was 1.56. Now the ratio of annuity to cash is 1.6. I've heard that it keeps going up as the jackpot increases. Why is this? Thank you.

• Author

TR Smith 22 months ago from Eastern Europe

The quoted cash value is the estimated amount of the jackpot on that day; the quoted annuity value is the estimated total of all the graduated annual payments based on what they estimate the returns will be when they invest the jackpot money. (If you choose the annuity option, the lottery invests the jackpot and pays you annually for 30 years. These payments increase by 4%-5% each year.) Exactly how they estimate these things is not public information.

• zan 22 months ago

Thanks again. Today the jackpot is 188 million (annuity) and 119.4 (cash), which has a ratio of 1.57, lower than before. I guess the ratios fluctuate based on how they recalculate the estimates every week and I shouldn't look too deeply into it.

• Wll 15 months ago

If people elect to receive the annuity...and their payments are based on the returns of the invested money, the states must have quite a portfolio considering the jackpot sizes...wonder if that total should be public just as the winners names are? Could you pick a state that has a lottery..research their lottery history...and post their potential investment capabilities? TY

• Author

TR Smith 15 months ago from Eastern Europe

Hi Will, thanks for the comment. It sounds like an interesting project to undertake, but I wouldn't know where to begin. If you'd like to investigate the flow of money, you should start with your state's lottery commission and see if someone there can tell you how they invest it to fund annuities. My guess is they are not required to make all this information public and so they don't.

Better yet, you could get a job at the lottery commission and learn how it all works from the inside.

• Pattyanne32 14 months ago

So, the powerball lottery prize is now up to \$700 Million. The news is reporting that the odds of winning increase with the amount of the lottery prize and/or the number of people playing. I don't understand that, since the number of possible combinations of numbers that can be chosen on a ticket or that could be "picked" by the machine that spits out the little balls do not change with the prize amount or the number of people playing. It seems to me that the only thing that should change is the possibility that you would be SHARING the prize because with more people playing the numbers there is a better chance that more than one person could pick the same combination of numbers. Otherwise, mathematically speaking, the chances of picking the winning numbers should remain the same?

• Author

TR Smith 14 months ago from Eastern Europe

Yes, you are correct, the only probability that increases with more players is the probability of multiple winners. And it doesn't matter how many roll overs have occurred before, it only depends on the number of people buying a ticket for the current drawing. Usually when the jackpot rolls over a dozen times people who don't normally buy tickets will buy one or two, so you get maybe 3 or 4 times the usual number of people playing. I have another article about this

https://hubpages.com/education/Probability-of-Spli...

I have also read some poorly written "news" articles about lotteries that state all kinds of wildly inaccurate and false information. For example, an ABC news report said that since 75% of winners were computer picks, you should let the computer pick your ticket numbers. What the writer failed to mention is that 75% of all tickets sold are computer picks, which means that neither manual nor computer picks have an advantage over the other.

• Syl 14 months ago

Since I don't buy lotto tix very often it feels like every time I buy one the structure of the lottery is different. Last time I bought a PB ticket there were fewer white balls and more red balls. I guess my question is why do they change it all the time? Is there a mathematical reason why changing it benefits the lottery company? I know they're not doing it for the benefit of the players.

I'm also trying to wrap my head around these multipliers. I've never bought one because they don't improve you odds of winning, they only increase your prize if you do win, but if the multipliers are 2, 3, 4 and 5 then why isn't the average 3.5? is it because they don't all have the same chance?

• flatlander mcsandhiller 14 months ago

illinois is a lesson in why ppl should always take the lump sum, the state hasnt been able to pay all those chumps who took annuity options in lotteries of yore.

• Author

TR Smith 14 months ago from Eastern Europe

Hi Syl,

You're right that the lotteries never change the structure for the benefit of players, I believe the main motivation is to confuse lottery players by offering games with worse odds than before but larger potential prizes. The current billion dollar jackpot of the Powerball proves that their marketing efforts to induce "lottery fever" do work and that making the odds worse doesn't really deter people from playing. The odds of winning today are almost exactly twice as bad as they were in 2005.

As for the multipliers, they are not distributed evenly, that is why the average multiplier value isn't 3.5. In Mega Millions, they have 15 balls to determine the multiplier value. Six of them are "5," four of them are "3," three of them are "4," and two of them are "2." When one of them is drawn at random, there's a much higher chance of getting a "5."

In Powerball they have 43 balls to determine the multiplier. Twenty-four of them are "2," thirteen are "3," three are "4," two are "5," and one is "10." The "10" is left out when the jackpot is more than \$150 million. Unlike the Mega Millions whose multipliers skew to the high end, the Powerball's multipliers skew low. Again, the uneven distribution is not well publicized for the purpose of confusing people, you have to read the fine print on the state lotto website to see how they distribute them.

• Julio 14 months ago

I stopped playing Powerball when the odds became worse than Mega Millions. \$2 for a lottery with odds worse than getting struck by lightning while getting bitten by a shark? No thanks. And there's a point at which the jackpot starts to become unappealing. 1.3 billion is more trouble than it's worth. I'd rather stake my chances on a couple hundred thousand from Texas Two Step. It's enough to pay the bills but not so much that every cousin is coming out of the woodwork to take a bite of your prize. The Texas lotto report woman says this will result in decreased sales in the long run because the astronomical odds are too much of a turn off. What do you think?

• Author

TR Smith 14 months ago from Eastern Europe

Hi Julio, thank you for the comment. I don't really understand lottery player motivation, but it wouldn't surprise me if they go back to a format with better odds because they can't get enough sales to sustain interest in this new version except during the rare huge jackpots that require dozens of rollovers.

• Julio 14 months ago

Thanks for your reply. I have another question: What is the least number of people who have to buy tickets to guarantee there'll to be a winner?

• Author

TR Smith 14 months ago from Eastern Europe

It's not the number of people but the number of unique combinations that are sold. If the lottery manages to sell at least one of every possible combination for a particular drawing, there will be at least one winner with 100% certainty. If only 30% of all the possible combos are sold then there's a 30% chance of having a winner and 70% chance of having a rollover.

• Shannon 14 months ago

I think you did a great Job with this article and answers. Here in WV I hear the cash 25 is one of the best lotteries in the US and has been around since 1990. 4 days a week 25 balls and you must get 6 of the 25 for top prize of 25 k but odds are around 177k. Already playing playing at about 1 or 2 dollars a drawing for about 8 draws I have broke almost even with a couple 3 and 4 ball winners. 3 gets you 1 dollar, 4 gets you 10 and 5 gets 250 with 6 of course the 25k. What's the break even point for this game as I read that about 85% of all combinations have already occurred and a bunch even multiple times. Also in this type of game is it better to play same numbers over and over as I am doing? Thanks

• Author

TR Smith 13 months ago from Eastern Europe

Hi Shannon, thanks for your comment. About break-even points in lotteries, unless the payout structure is flawed or the number selection is non-random in a predictable way, there shouldn't be any break-even point. Anyone who plays the lottery enough times will lose more than they earn.

For example, in WV Cash 25, the expected return per \$1 spent is about \$0.556352. In concrete terms, if a hundred people played Cash 25 a hundred times each, they would spend \$10,000 but their winnings would most likely only amount to about \$5,563.52.

As for what numbers you should play -- the same numbers over and over, or different numbers each time -- as long as the lottery is using a random process then it doesn't make any difference. But if there's some flaw in the drawing process and certain numbers are unlikely to be picked, then it makes sense to avoid those numbers. The problem is you usually can't tell if a game is flawed. Even if the frequencies of certain numbers are relatively high or low, it might not be statistically significant enough to work it to your advantage. Lotteries usually keep an eye on these things, but there have been cases of flawed games.

• Eric 2 months ago

Even though you are losing money, buying a ticket for a buck can be worth it because it can get you through a boring week.