How to Win Powerball According to Math

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Last updated on May 25, 2024

Are you curious about how to win Powerball? If so, let me introduce a powerful strategy based on combinatorial math and probability theory. Get ready to gain a favorable advantage despite Powerball’s inherent randomness.

My analysis of the Powerball game shows that one group of combinations exhibits a favorable success-to-failure ratio despite all combinations having an equal probability of winning.

First, we must understand how the game works. Then, we will establish a realistic strategy and make informed choices based on what is mathematically possible within the game’s mechanics.

If that piques your interest, let’s begin.

The Odds of Winning in Powerball

The Powerball game follows a 5/69 lottery format, which means you must choose five from a set of 69 numbers. There are a total of 11,238,513 possible ways to combine five numbers.

To win the jackpot, you must also match the extra ball called the red Powerball from 1 to 26. So the probability of winning if you buy one ticket is 1 in 292,201,338.

The odds are so terrible that you probably have a better chance of becoming the next Governor of the State of California. So it’s not easy to win. If Powerball is one of your favorite lottery games, then you might as well do your best shot.

Aside from the jackpot, you also win eight more likely prizes ranging from $4 to $1 million. So, based on the odds and prizes of the Powerball game, the average odds of winning a prize are 1 in 24.87.

Therefore, the probability that a single ticket isn’t a prizewinner is 0.9598, and the probability that you lose nth times is this number raised to the power of n. For example, if you buy two tickets, the probability of losing is 0.9212.

P(losing twice) = 0.95982 tickets

Consequently, to get a 50/50 chance of winning any prize, you must buy 17 tickets. We calculate this using the complementary of P(losing). Thus:

P(winning any prize) = 1 – 0.959817 tickets = 0.5022

To attain a 99.99% certainty of winning a prize, you should buy 224 tickets.

Please be aware that this 99.99% guarantee most likely refers to winning the $4 payout, as the probability tends to favor the lowest-tier prize.

This image shows how buying more Powerball tickets increases your probability of winning. Buying 17 tickets will give you a 50/50 chance. While 224 tickets will give you 99.99% of winning any prize, more likely lowest-tier-prize.
This chart shows two lines. The blue line representing the winning probability and the red line representing the losing probability. The two lines intersect at 17 tickets indicating the point where Powerball players get a 50/50 chance of winning any prize by buying 17 tickets. Although the potential win could be the lowest tier prize.

As you might have noticed, winning the smallest prizes in Powerball, not to mention the jackpot, is quite a challenging task. Therefore, if anyone sells a strategy that promises consistent profit from the lottery, please run away. Claims like that are misleading due to the game’s negative expected value, which means, on average, you’re more likely to lose money than win.

Understanding Powerball’s Expected Value

We emphasize purchasing more tickets as the main strategy for increasing your probability of winning. However, the expected value is another important factor when playing Powerball.

What is the expected value?

In the context of the Powerball lottery, the expected value or EV is a mathematical concept that represents the average amount of money a player can expect to win (or lose) per ticket purchased, taking into account all possible outcomes, their probabilities, and their respective prizes.

The Powerball 5/69 showing a negative expected value of -0.37

The table indicates that the expected value (EV) is negative, suggesting that playing Powerball is not likely to be profitable when the jackpot prize is $384 million.

In contrast, when the U.S. Powerball jackpot reached $1.5 billion in January 2016, there was a significant increase in the expected values, indicating a better potential for profitability.

The Powerball 5/69 showing a positive expected value of $3.45 at 1.5 billion jackpot prize level

Typically, due to the high probability of not winning anything, the expected value of a lottery ticket is always less than the ticket price. However, the expected value can exceed the ticket’s cost when the jackpot prize reaches exceptionally high. However, it’s important to remember that the odds are designed so that the expected value will generally be negative, reflecting a loss on average for players.1

Playing the Powerball With a Mathematical Strategy

You have likely heard that a shark attack is more likely to kill you than winning the lottery. However, realize that if you never swim in the ocean, your risk of a shark attack is not probabilistically possible.

The same holds for winning the Powerball. You’ve got to be in it to win it.

In short, you must buy a ticket to have a chance of winning, and buying multiple tickets is the only mathematical way to increase your odds.

But buying multiple tickets is useless if you’re not making the right choices.

Take a look at the following combinations:

10-20-30-40-50all numbers ending in zero
11-22-33-44-55skip counting by eleven
29-39-49-59-69all numbers ending in nine
1-11-12-21-22digits limited to one and two
2-12-22-32-42all numbers ending in two

Let me ask: Are you willing to buy Powerball tickets using all previously mentioned combinations?

I asked a couple of lottery players, and surprisingly, none were willing to take a risk.

That begs the question, why?

Why hesitate when they believe all combinations have an equal chance of winning? Either they don’t trust their math2 or their belief in math is based on a weak foundation.3

Sure, gut feeling saves you from wasting your money, but the same gut feeling prevents you from winning.

The point is that your gut feeling makes more sense when supported by calculated reasoning.

There must be a mathematical reason you feel good about some combinations and avoid others when buying lotto tickets.4

For one, we must ground every decision in solid mathematical logic to make informed decisions.

Statistics will not help.

Statistics is not the right tool for lottery games.

To explain briefly, a lottery game is finite. We don’t need statistical sampling to understand the probability of many events in the lottery. We have sufficient data to infer the game’s composition when dealing with a finite structure.

In the context of the lottery game, we use combinatorial mathematics5 and probability theory6 to understand how balls behave in a random game. These two mathematical tools will help calculate the many possible outcomes of the game and reveal certain information that will help us make intelligent choices when playing the lottery.

Lottery players get this information in the form of a success-to-failure ratio.

Decoding the Science of Success and Failure in Powerball

In a truly random game, all numbers and combinations share the same probability of winning.

So, how do you make informed choices when all the numbers and combinations have the same probability?

Well, the key is the concept of the success-to-failure ratio.

To fully grasp this all-important ratio, realize that odds and probability are two different terms. They are not mathematically equivalent.7

We express probability using this formula:

P(success) = all favorable events / all possible outcomes

Here, we can say that P(success) is complementary to P(failure)

P(failure) = 1 – P(success)

The probability formula measures how likely an event is to happen, while odds compare the number of ways an event can succeed and the number of ways it can fail.

Odds of winning = P(success) / P(failure)

So, odds provide a clearer picture of your advantage. In Lotterycodex, we call this your success-to-failure ratio.

A lottery strategy exists because combinations have varying compositions, exhibiting different success-to-failure ratios. You should understand how this ratio can improve your number selection strategy in Powerball.

How do you make informed choices?

So, even though you cannot change the underlying probability and you cannot beat the lottery’s odds, you have the power to calculate possible outcomes of a random game and make intelligent choices based on the success-to-failure ratio that is most favorable to you.

So, the strategy is to choose a success-to-failure that works for you.

Allow me to explain by comparing two combinatorial groups. One group comprises 5-even numbers, while the other is a well-balanced composition of 3-odd and 2-even numbers.

There are 278,256 ways to combine 5-even numbers in Powerball. This number suggests that a composition of 5-even numbers should occur approximately twice in every 100 draws.

Odds(0-odd-5-even) = 278,256 / 10,960,257 = 1/39.39

The 1:39 ratio means that out of 40 draws, you will have one favorable shot and 39 unfavorable ones.

As a lotto player, you probably wouldn’t want to spend your money on 40 draws for only one favorable shot.

On the other hand, there are 3,671,745 ways to combine 3-odd and 2-even numbers in Powerball. So, this group will occur about 33 times in 100 draws.

Odds(3-odd-2-even) = 278,256 / 7,566,768 = 1/2

A success-to-failure ratio of 1:2 indicates that you get 33 favorable shots in 100 draws.

How do we explain this random behavior of the lottery?

According to probability theory, if you divide the number field into odd and even sets, a truly random game spreads the probability fairly between the two sets. Therefore, drawing a winning 6-even combination is rare. Since the game spreads the probability evenly between two sets, the most dominant combinations will be 3-odd and 2-even.

Let’s compare the two compositions to see the big difference:

As you can see from the table above, there’s a mathematical reason why you should pick one and avoid the other. As a lotto player, you must choose the one that will give you the most favorable shots. Usually, these are the dominant compositions in a lottery game.

As a Powerball player, you have the power to choose a success-to-failure ratio that works for you. You cannot change the underlying probability and you cannot beat the lottery’s odds, but you have the power to calculate possible outcomes and make an informed choice. Even choosing not to buy a Powerball ticket is a strategy by itself if you think the odds are too monumental.

The Powerball Game Follows the Dictate of Probability

The game’s outcomes will exhibit a deterministic behavior when looking at many lottery draws collectively. This happens because the law of large numbers must take effect despite each draw having an independent probability. The law of large numbers states that the game’s outcome will generally follow an expected trend based on probability theory as more draws occur.8

The image of randomness shows streaks and clusters.
Streaks and clusters describe a truly random lottery game. This random behavior provides insights for more strategic number selection. Learn more: A Truly Random Lottery with a Deterministic Outcome.

In mathematics, we compute the expected frequency of each combinatorial group by multiplying the probability by the number of draws. Calculating probability estimation is an important aspect of learning how to win Powerball.

Expected Frequency = Probability X number of draws

From October 7, 2015, to March 16, 2024, there were 1,008 draws in Powerball. So, in the case of a 3-odd-2-even pattern, we get 329 by multiplying 0.326710926970499 by 1,008.

Expected frequency (3-odd-2-even)
= 0.326710926970499 x 1,008
= 329

The 3-odd-2-even composition is expected to occur about 329 times in 1,008 draws.

By performing similar computations with the rest of the odd-even compositions, we arrive at the following complete comparison graph below:

This is the US Powerball odd-even analysis updated as of March 16, 2024 with 646 draws.  The graph shows close agreement between probability prediction and actual draws.

As you may notice, the close agreement between expected and actual frequency proves that Powerball behaves in a predictable trend.

The drawing of the Powerball game follows the dictate of probability theory.

As a lotto player, you can take advantage of this probability principle to play the Powerball and be wrongless for most of the draws. Thanks to mathematics.

How to Choose the Best Powerball Numbers

Probability analysis can be problematic if not done correctly. If you employ the wrong method, it can lead to a misleading and inaccurate conclusion.

Let me explain.

Based on our discussion above, a composition of 3-odd and 2-even exhibits a dominant characteristic and thus might be a good choice for Powerball players.

But this couldn’t be further from the truth.

Probabilistically speaking, a combination composed of purely low numbers cannot have a favorable success-to-failure ratio.

Not all combinations in the 3-odd-2-even group exhibit a favorable success-to-failure ratio.

How do we produce an accurate analysis?

We have to define sets of numbers that will equally spread the probability across the number field, ensuring that odd/even and low/high numbers are considered.

We designed Lotterycodex to represent the random behavior of the Powerball game accurately. Hence, we group the 69 numbers into four sets:

Powerball 5-69 Lotterycodex combinatorial design

Generated by Lotterycodex Calculator

These numbers used in the Powerball game contain complex compositions that enable players to distinguish dominant groups from rare ones easily.

In Lotterycodex, we describe these compositions as templates that serve as simple guides, delivering the mathematical strategy on a silver platter. It would help if you started familiarizing yourself with these templates when choosing numbers to play Powerball.

Generated by Lotterycodex Calculator

As a lotto player, you don’t want to spend your money on a composition expected to occur only thrice in 5000 draws. An example of this combination belongs to the group of template #56.

If you choose numbers randomly without any probability guide, you might unknowingly select a combination that belongs to the rare groups without realizing it.

How To Win Powerball

I always recommend that Powerball players focus on the dominant groups since they exhibit the most favorable success-to-failure ratio.

If you don’t like calculating, you can use a Lotterycodex calculator to do combinatorial and probability analysis. You can use this calculator to generate combinations based on your covering set.

There is only one way to increase your odds of winning Powerball. You have to buy more tickets. Purchasing multiple tickets can be done in two ways:

  1. The basic method is choosing numbers randomly or using a quick pick machine.
  2. A more effective method is to use a lottery wheel. This technique methodically traps the winning numbers within your selection of numbers. This strategy doesn’t happen in the basic method.

I recommend using a lottery wheel. Learn more: Lottery Wheel: A Clever Mathematical Strategy That Works.

One of the best features of the Lotterycodex calculator is that you can use it as a lottery wheel. At the same time, it separates combinatorial groups based on their corresponding success-to-failure ratio, allowing you to make intelligent choices when playing the Powerball game. Read on: Lottery Calculator: A Mathematical Guide Beyond Number Selection

Join the Powerball Conversation

Do you know any tips on how to win Powerball? Let me know your thoughts. Join the conversation by adding your comments below. Here are some questions and answers that might help start the conversation.

What are the odds of winning the Powerball jackpot with a single ticket?

The Powerball game uses a 5/69 format, resulting in 11,238,513 possible combinations of five numbers. To win the jackpot, you must also match an additional ball drawn from a range of 1 to 26. Therefore, the probability of winning the jackpot with a single ticket is 1 in 292,201,338.

How does buying more tickets impact your winning odds?

Buying multiple tickets is the only mathematical way to increase your odds of winning, but it’s crucial to make informed choices about the combinations you select. Since combinations have varying success-to-failure ratios, I advise Powerball players to choose combinatorial groups that dominate the Powerball draw over time to get the best shot possible. The cost of buying more tickets adds up quickly, so players should consider participating as a group. Using a lottery wheel by a lottery syndicate can effectively enhance the strategy of buying more tickets.

What role does the success-to-failure ratio play in Powerball?

The success-to-failure ratio in Powerball is a crucial concept that helps players understand their chances of winning versus losing. This ratio measures the number of favorable outcomes against the number of unfavorable ones. It is a simple guide for making more informed decisions when choosing number combinations instead of relying on random picks or superstition-based strategies.

Do statistics help Powerball players?

The Powerball game has a finite structure; therefore, any questions we pose about the game are probability problems to solve rather than statistical ones. Combinatorial mathematics and probability theory are the right mathematical tools to understand how the balls behave in a random lottery draw. These tools help calculate the many possible outcomes of the game and reveal certain information that assists Powerball players in making intelligent choices. For players looking for tips on how to win Powerball, Lotterycodex presents this information in the form of a success-to-failure ratio.

Can mathematical strategies guarantee a Powerball win?

While mathematical strategies can enhance the selection process by identifying combinatorial groups with more favorable success-to-failure ratios, they cannot alter the game’s inherent randomness and low probability. Powerball and similar lottery games are designed to make each draw an independent event, making the outcome uncertain and fair for everyone.

Thank you for reading. :)

Additional Resources

  1. Expected Value
    According to math, it’s not worth risking $2 to play for the $1.5 billion Powerball jackpot
        []
  2. When to Trust Your Gut    []
  3. Developing Your Intuition For Math    []
  4. Do The Math, Then Burn The Math and Go With Your Gut    []
  5. Combinatorics    []
  6. Probability Theory    []
  7. What is the difference between odds and probability?    []
  8. Law of Large Numbers    []

26 comments

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  • You’re right 3 odds and 2 even 3 even and 2 odds , 2 high and 3 low or 3 low and 2 high… Even the tens are applied. I didn’t win the jackpot yet but I missed one time to buy the ticket, the next morning I checked the numbers for curiosity , and I found out the 5 white balls were all my numbers ! But I missed to buy the ticket ! :( I used your strategy. So it is proven this strategy works !

  • I’m novice and just started to explore how to play Power ball lottery game and curious to the Statics behind this game.
    your presentation with mathametical combinations and the way you explained everything is really very helpful for me.
    thank you very much, sir.
    thanks Best regards 😊
    R.P.Singh

  • Interesting you didn’t mention using birthdays and other dates. Since birthdays are confined to 31 days, and twelve months, it means that if any of these days come up as winners there’s a greater likelihood of more winners, and therefore smaller, shared jackpots.

    Same with people who use “lucky” numbers like 7, and famous sequences like 4, 8, 15, 16, 23, 32 which were the “LOST” numbers from the TV series. I can imagine there are thousands of people playing those numbers, and if they were to come up the shared jackpot would be minimal.

    • The LOST TV show lottery numbers did come up in June 2011 in the Mega Millions lottery. There were 41,763 people who played the numbers and each of them won $150

  • I thought your article was interesting. I am not very receptive to all the statistics but willing to try to learn more and understand more of what you are saying. Keep up the good articles.

  • A wonderful Read, Knowing this i can combine the information with the information of the past winning numbers to further reduce the probability of losing a draw, i also learned how to use a calculator to get the possible combinations of any real life appliance.

    • When dealing with a finite game like a lottery, past winning numbers are unnecessary. In a finite game, any questions you ask are probability problems to solve rather than statistical. Therefore, there is no need to analyze past winning numbers.

  • Great information; lot to read lol. Thank you for taking the time to study and explain this. It’s worth a try!

  • This was truly interesting and not a waste of time unlike others that I’ve read. Makes some sense although I do believe to some point there is luck involved. Thank you so much for sharing!!!!

  • Thank you ,this was a breath of fresh air for me, I spend hundreds of dollars trying to catch both games I try everything, anyway I was able to catch 4 numbers in both games already, in the power and in the mega, I,m 63 yers and I want to win, and to lived a comfortable life after I win, may god help me, I learn a lot from this lay out god bless you and put some luck on me.

  • Thank you for an interesting insight to one of my poorer subjects Maths. I think every Number has an equal opportunity as does 12345 and 6 for Lottery games in Australia. I’ve played and won various prizes along way. My theory to how I play is simple, although odds remain the same. EG I play 20 games selecting 6 numbers from the available 45. But in all the 20 games, 2 numbers remain in all 20 games. If those 2 numbers drop in the draw I believe it makes it slightly easier to match the remaining 4 random. I’ve had a little success over the years. But never the grand prize. Nevertheless it’s a bit of fun. That’s the way you should treat it. Might try the crystal ball next Thank you again

  • How can i find a Lottery Codex Calculator ? ,,, i bought a toy ( made in China ) but it has up to 30’s #s , no 40’s , 50’s or 60’s as PB and MM play . Thanks for your help . ( i’ve try several methods in the past w/o positive results )

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