How to Win Mega Millions According To Math


Last updated on June 7, 2024

Knowing how to win Mega Millions requires understanding combinatorics and probability theory. My examination of the game reveals that, out of 56 groups, two exhibit a notably high success-to-failure ratio, even though each combination holds an equal chance of winning.

This article examines a math strategy focusing on S/F ratios. First, we must understand the challenges and then discuss how to overcome them within the game’s framework. Of course, the fun starts with your number selection strategy, and your job is to make informed choices to get the best shot possible.

When you’re ready to learn, let’s get started.

The Probability of Winning the Mega Millions

The Mega Millions 5/70 game has 12,103,014 playable combinations. Therefore, the probability of matching five correct numbers is 1 in 12 million chances. But to win the Mega Millions jackpot, you must match the extra mega ball by matching another number from 1 to 25. This requirement pushes the probability to a monumental 1 in 302.5 million chances.

You cannot win the jackpot without buying a ticket, so my first tip on how to win Mega Millions is to be in it to win it.

Fortunately, besides the jackpot, you can also win eight more likely prizes ranging from $2 to $1 million. So, on average, players get a 1 in 24 chance of winning any prize.1

Winning a five-number match without the mega ball is much better, with a 1 in 13 million chance. If you win the second prize, you get one million, which is life-changing.

It’s not easy to win.

The probability of an individual ticket not winning any prize is 0.9583. Consequently, the chances of experiencing losses by buying nth tickets are this number raised to the power of n. For example, the probability that you lose twice in a row is 92%.

P(losing twice) = 0.95832 tickets = 0.9184

Therefore, you must purchase at least 16 tickets to achieve a 50/50 shot at winning any prize. Yet, there is no assurance. To have a 99.99% certainty of winning a prize, you must buy 216 tickets. We determine this by calculating the complementary of P(losing) as shown below.

P(winning any prize) = 1 – 0.9583216 tickets = 0.9999

It’s important to understand that the odds favor securing the lowest-tier prize. In this case, you are most likely to win $2.

This calculation reinforces that you cannot profit from playing Mega Millions game. Therefore, don’t believe anyone who offers you a strategy that promises frequent small wins while you wait to win the jackpot. The lottery doesn’t work this way.

Playing Mega Millions with Mathematical Strategy

Buying more tickets is the only way to increase your winning chances. However, due to the lottery’s negative expected value, buying more tickets tends to be expensive in the long run.

And buying a hundred tickets can be useless if you make the wrong choices. For one, you don’t want to spend your money on tickets with the following combinations:

4-5-6-7-8straight consecutive
10-20-30-40-50multiples of 10
1-11-21-31-41all numbers ending in 1
5-10-15-20-25skip counting by 5
3-9-15-21-27all odd numbers and skip counting by 6

These combinations above are not the most sensible choices you can make for yourself as a Mega Millions player.

You might want to consider playing with groups of combinations that frequently occur. The last thing you want to do for yourself is to spend money on a lotto ticket that will only give you one favorable shot after playing 10,000 draws.2

However, some people insist that all combinations have the same probability. And I do agree with this belief because there’s only one way to win the jackpot.

P(winning the jackpot) = 1 / 302,575,350

However, strategizing your play involves looking at the game in a different light.

Let’s consider the combinations 10-20-30-40-50 and 11-22-33-44-55.

Some players will hesitate to take a risk because “gut feeling” is much stronger than their understanding of math.

You see, we have to explore a solid mathematical logic as to why we feel good about some combinations and fret with others.

That’s the power of mathematics: You can make informed choices, and thus, you don’t rely on gut feeling when making decisions.

Minding Your Success-to-Failure Ratio

While all combinations have the same probability, combinations may not be created equally. Combinations have different compositions. For example, 1-2-3-4-5 is a combination of low numbers. 1-3-7-11-15 is a combination of odd numbers. And 10-12-14-16-18 is a combination composed of purely even numbers.

Since combinations have different compositions, different combinatorial groups with varying success-to-failure ratios exist.

What is a success-to-failure ratio, and why is it so important?

In mathematics, odds and probability are two different terms. They are not mathematically equivalent.3

We use probability to measure how likely an event will occur. We use the following formula:

Probability is the number of favorable combinations over the number of total combinations.

And we use the odds to compare the number of ways you get favorable shots against the number of ways you don’t. We use this formula below instead:

Odds compare the number of favorable events with the number of ways you don't get favorable events.

As a lottery player, you must ensure you get more favorable shots. So, the calculation of odds reflects the success-to-failure ratio.

How do we interpret this S/F ratio?

For example, there are 324,632 ways to combine purely five-even numbers. Let’s call this group the 0-odd-5-even group. There are 12,103,014 playable combinations in Mega Millions, so 12,103,014 minus 324,632 gives you 11,778,382 ways you don’t get favorable shots.

Now, there are 3,894,275 ways to combine 3-odd and 2-even numbers. So, the number of ways you don’t get favorable shots is 8,208,739.

According to the probability theory, a truly random lottery game draws the winning combinations more frequently from the 3-odd-2-even group.

The dominance of the 3-odd-2-even group occurs because if you divide the numbers into odd and even sets, a truly random lottery game distributes the probability fairly and evenly across odd and even numbers. Therefore, a random drawing will neither favor the even group nor the odd group.

As a result, it’s very rare for a lottery game to draw a combination composed of purely even numbers or odd numbers.

Now, we use the success-to-failure ratio to explain this behavior mathematically. Take a look at the comparison table below:

To win the Mega Millions, you have to ensure that you are choosing odd and even numbers properly. This image shows that 3-odd-2-even composition has a better success-to-failure ratio than 0-odd-5-even composition.

Calculated using Lotterycodex Calculator

Mathematically speaking, the 0-odd-5-even group will give you 32 favorable shots in 100 draws. The ratio of 1:2 means that you get one favorable shot in every three attempts, which is better than the 0-odd-5-even group.

As a Mega Millions player, you don’t want to spend your money on 37 draws just to get one favorable shot. Do you?

Mega Millions Draw Follows the Dictate of Probability

At the time of writing, there were only 409 draws to analyze as of November 12, 2021. Even with that limited number of draws, we can say it’s big enough to come up with a conclusive comparison between the actual results and the probability prediction.

Probability calculations always agree with the actual results.

Below are the graphs we have prepared to show the agreement between probability prediction and actual Mega Millions draws. The data were from October 31, 2017, to March 8, 2024, which composed of all the winning combinations from 628 draws:

The Mega MIllions odd-even analysis updated as of March 8, 2024 with 628 draws.  The estimated frequency closely matched that of the actual frequency which proves that the results of the US Mega Millions are always subordinate to the dictate of the probability theory.

Using the probability theory, we can predict the likely outcome of the Mega Millions game to an extent. For example, we can predict the group that will dominate the lottery draw based on the law of large numbers.

How to Pick the Best Numbers in Mega Millions

There’s more to Mega Millions than odd and even numbers.

Probability analysis can be confusing if you’re not careful with your calculation method.

For example, 1-2-3-4-5 is considered dominant under odd/even analysis. Yet, such a straight combination of low numbers will not give you a favorable success-to-failure ratio.

Lotterycodex combines odd, even, low, and high numbers into a single analysis to ensure accuracy. For more details, please read The Winning Lottery Formula.

Lotterycodex Analysis of the Mega Millions

Here’s the set of numbers for the Mega Millions under the Lotterycodex analysis:

This image shows the Lotterycodex Combinatorial sets for a 5/70 game. The number field is divided into LOW-ODD, LOW-EVEN, HIGH-ODD, and HIGH-EVEN sets.

Generated by Lotterycodex Calculator

Looking deeply into the finite number field of a 5/70 lottery game, you will discover layers of combinatorial templates that will give you a more favorable success-to-failure ratio.

For example, 7-10-37-55-67 combines one number from LOW-ODD, one from LOW-EVEN, and three from HIGH-ODD sets. In Lotterycodex, we call it template #27. This combination belongs to the group with a very low success-to-failure ratio of 1:57. This ratio means you get only two favorable shots in 100 draws.

As a Mega Millions player, you don’t want to spend your money on template #27 to get two favorable shots in 100 attempts.

My probability study shows that of 56 templates, only two are dominant compositions in the Mega Millions game.

According to Lotterycodex analysis, the Mega Millions have 56 templates. Of 56 templates, only two are dominant. These are templates #1 and #2.

Generated by Lotterycodex Calculator

If you pick numbers randomly without any combinatorial and probability guide, your combination might fall into one of the rare groups in Mega Millions, and you don’t even know it.

Compositions with a Poor S/F Ratio Abound

Take a look at these combinations below:

badly constructed combinations in Mega Millions lotto game

The combinations above are examples of combinations with a poor success-to-failure ratio.

Your objective is to win the Mega Millions game. You aim for groups that will give you more favorable shots. Lotterycodex Calculator will help you make informed choices using the success-to-failure ratio. We use combinatorial4 and probability theory to achieve this objective.5

How To Win Mega Millions Game

Play the Mega Millions game for fun. But do it with a strategy using the Lotterycodex calculator, and playing will be much more fun.

Remember, buying more tickets is the only way to increase your chance of winning. This strategy can be more effective using a lottery wheel rather than buying random tickets. This is because, mathematically speaking, a lottery wheel strategically traps the winning numbers. This strategy cannot be done when you buy tickets randomly.

Lotterycodex created the Lotterycodex calculator to provide a more effective lottery wheeling system that separates the dominant group so you can make better decisions when picking numbers.

Predicting the behavior of the Game

The image below shows you how the Mega Millions game will behave over time:

Using Lotterycodex analysis, Templates #1, and #2 will dominate the Mega Millions draw and will continue to dominate as more draws take place according to the law of large numbers.

Generated by Lotterycodex Calculator

Focus on the dominant groups (templates #1, #2) to get the best shot possible. According to the law of large numbers, templates #1 and #2 will continue dominating as more draws occur.

Avoid template #55, as this template occurs only three times after 5,000 draws.

You don’t want to spend money on 5000 draws to get only three favorable shots. Do you?

These Lotterycodex templates will help you make informed choices. Math remains a powerful guide when selecting numbers.

While learning how to win Mega Millions, please play the lottery responsibly. Implement a lottery game plan to keep your spending going beyond your means.

Moreover, anyone could be at risk of the lottery addiction problem. Like I said earlier, the lottery is just a fun game. Your gambling losses are merely the price of a good time, much like cinema tickets are the cost of the entertainment.

Questions and Answers for Mega Millions

What is the probability of winning the Mega Millions jackpot?

The probability of winning the Mega Millions jackpot is extremely low, with odds of 1 in 302,575,350. The players must match five numbers from a pool of 70 balls and one additional number from a separate pool of 25. The immense number of possible combinations makes the jackpot very difficult to win.

How can one guarantee winning any prize in Mega Millions?

Buying 216 tickets might suggest a 99.99% chance of winning a prize in Mega Millions, but given the odds, the likelihood of winning lower-tier prizes is higher than bagging the jackpot. However, it’s worth noting that the expected value of playing the lottery remains negative, implying you’re likely to spend more on tickets than actually winning the game on average. Since each draw is independent and no strategy can change the underlying odds, it’s crucial to play responsibly, recognizing the lottery as entertainment rather than a reliable source of income.

How many Mega Millions tickets should one buy for a 50/50 shot at any prize?

To have a 50/50 chance of winning any prize in Mega Millions, you must buy at least 16 tickets. However, given the odds, you will likely win the lowest-tier prize. Fortunately, it’s not impossible to win the jackpot as long as you purchase one ticket, although chances are astronomically low.

Does buying more tickets improve Mega Millions jackpot odds?

Yes. Purchasing more tickets can certainly increase your chances of winning the jackpot. However, this approach can be costly and less effective without a strategy. A recommended strategy is using a lottery wheel to strategically trap more winning numbers and play as a syndicate to mitigate the cost of playing.

How does the Lotterycodex calculator help choose Mega Millions numbers?

The Lotterycodex calculator aids number selection by evaluating and categorizing combinatorial groups into templates based on their success-to-failure ratios. These templates guide players in making well-informed choices.

Your Insights Would be Valuable

Do you know any tips on how to win Mega Millions? If so, let us know. Share your knowledge and experiences.

Thank you for reading. :)

Additional Resources

  1. Nine Ways to Win a Prize in Mega Millions    []
  2. Law of Truly Large Numbers    []
  3. What is the difference between odds and probability?    []
  4. Combinatorics    []
  5. Probability Theory    []

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