Should You Play Powerball or Mega Millions? Probability & Expected Return
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.
Become a Lottery Math Expert: Must Read Articles
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 lowertier 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 10valued 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 nonjackpot 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 breakeven 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 without votingOverall odds of not losing money: The overall odds for winning any prize, including a breakeven 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 secondtier prizes from the calculations, since you can consider the secondtier 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 multistate 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 selfstyled 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 biofeedback device that generates winning numbers based on your "body energy." Both items are pictured below.
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Comments 29 comments
so basically, it's a sucker's bet either way.
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!
Thanks, this is very useful and wellexplained.
Hello CG,
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/KenoLotteryStrategy...
Thanks for your help.
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?
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.
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.
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!
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.
Thanks for your answer to my previous question.
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.
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.
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
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?
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?
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.
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?
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?
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
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