# Should You Play Lotteries Based On Past Draw Results?

Did you ever play the lotteries based on past draw results?

If you did or continue to do so, you are not alone.

There are players who use exact winning numbers from previous draws.

Several players, meanwhile, often change numbers also depending on past draw results.

There might be a lot of players who follow these strategies.

However, popularity does not make the strategies effective and worth following.

An effective lottery strategy should be based on how lotteries work.

Mathematical concepts, instead of cognitive bias, govern the realm of lotteries.

Therefore, playing the lottery will be more worthwhile by applying mathematical analysis in your game.

Discover below how to avoid mistaken beliefs by favoring accurate computations and analysis.

# The Fallacies In Many Players’ Strategy

Researchers found a clear and consistent decrease in the amount of money placed on certain numbers after they appeared in lottery draws. One explanation researchers provided for this pattern is Gambler’s Fallacy. The decrease in betting on past winning numbers resulted from the widespread misconception among players.

Many players believe that a number drawn today has low probability of being drawn again in the following weeks/months. Thus, they switch to other numbers or stop playing until they believe the numbers will show up again.

Meanwhile, other researchers observed that past winning numbers had more bets on the succeeding draws. This shows that players also fall prey on the Hot Hand Fallacy. Influenced by this fallacy, players suppose that some numbers have a lucky streak that is likely to continue. So, they attempt to benefit from this by using the numbers in subsequent draws.

Do you also have these misconceptions? If you do, keep reading below to discover if they can help achieve your goal to win.

# The Problem With Fallacies

These fallacies are cognitive biases wherein you disregard the independence of chance events like lotteries. In lotteries, past and future events are independent from one another. Thus, the past draw results have nothing to do with future draw results.

As a lottery player, it is important that you consider every draw as independent. This will significantly help reduce the fallacies. Looking at past draw results might inspire or show what numbers to pick.

Yet, acknowledge that following these strategies will cripple your chances of winning. There could be more players like you, who follow the same method of playing. Therefore, you might be sharing the jackpot with many players, if ever you win.

Falling prey to these fallacies also lead to unnecessary waste of money. You assume you will win one day using a combination that might have the lowest odds of winning.

Remember that lotteries work according to specific laws of mathematics. Hence, you must use appropriate mathematical concepts if you want to play successfully.

# Law Of Small Numbers Vs Law Of Large Numbers

Some players pick the same set of numbers in all their attempts. Others change their numbers frequently depending on the results of lottery draws.

These players act according to the law of small numbers. Now, do not be quick in saying that this law of small numbers is a mathematical concept just because of the word “number”.

The law of small numbers is actually an incorrect belief by laypeople and experts alike. This is an assumption that small samples resemble the greater population.

The way our cognitive systems work allow this systematic deviation from reality to occur. This explains the tendency of people to create conclusions from short sequences of outcomes. They suppose these conclusions will apply to longer and bigger outcomes.

Applied in lotteries, this law of small numbers makes players believe that their observation from a few numbers of draws applies for more draws in the future. The strategies based on the gambler’s and hot hand fallacies could also reflect the use of statistics.

There are players looking at past results to determine hot and cold numbers. In lotteries, there are no hot and cold numbers. Instead of the law of small numbers, the law of large numbers governs the lotteries.

This law of large numbers is more applicable to lotteries that have hundreds and thousands of draws. It says that with many draws, all numbers will converge at the same frequency of occurrences.

By understanding this law, you will be able to create a strategy based on combinatorics and probability.

This image above illustrates the random, but deterministic nature of lotteries. There is no way to know what the next winning numbers will be. Yet, the appropriate mathematical tools allow a perceptive player to see the lottery game’s future deterministic behavior.

The white and gray specks in this image above are the combinations of a lottery game. They suggest how you can play the lottery successfully. Mind you, this suggestion is for playing lottery games by not being mathematically wrong. It is not about knowing the next winning combinations. (Find Out more by reading Free Guide: The Winning Lottery Formula Based on Combinatorics and Probability Theory)

# How To Not Be Mathematically Wrong In Playing Lotteries

The key in accomplishing this goal is to choose the appropriate math tools. There are different mathematical tools that players use, but are they all suitable for lotteries?

Statistics for hot and cold numbers, for example, is a tool commonly used by many players. To use statistics, players look at results from past draws (often 20–100).

A few draws may show frequently and less frequently drawn numbers. Yet, this observation will change when the number of draws increases. Thus, statistics is not a suitable lottery tool.

A lottery combination comprises the required amount of numbers per game. A 5/52 lottery game, for example, requires 5 numbers picked from 1 to 52. This game has 2,598,960 possible combinations.

These finite characteristics also make statistics an inappropriate lottery tool. So, instead of statistics, probability would be more helpful.

The combination you use must match the winning combination to win. Probability lets you know how likely you will win in a lottery game. If you bought a ticket with the combination 1–2–3–4–5 for the 5/52 game, your probability of winning is 1 in 2,598,960.

Each of the remaining 2,598,959 combinations like 48–49–50–51–52 and 16–22–25–25–52 offers the same probability. Let me remind you, however, these are only sample combinations and not recommended combinations to use in the actual game.

So, you can increase your winning probability by playing more lines. It is understandable you would not mind what combination to use because of the equal probabilities.

Not picking the most favorable combinations is a waste of time and money. Considering another related mathematical concept, therefore, is necessary.

If you compare the favorable combinations with unfavorable combinations, you would know the odds or ratio of success to failure.

Using odds, you discover that not all combinations are the same because their ratios are unequal. The odd, even, low and high number composition gives each combinatorial group a unique ratio.

To not be mathematically wrong in most draws, it is crucial to pick combinations with best ratios. You need to perform basic combinatorial analysis to determine a combination’s ratio of success to failure.

# Which Combinations Have The Best Odd-Even Ratio?

In a 5/52 game,

The odd and even number sets are

Odd = {1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51}

Even = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52}

To play this game, buy a ticket for at least one combination with any of the patterns below.

1. 5-even/5-odd

Each of these two patterns offers 65,780 ways of winning and 2,533,180 ways to lose.

2. 1-odd-4-even/4-odd-1-even

There are 388,700 ways to win and 2,210,260 ways to fail from either combinatorial pattern. Using 1-odd-4-even/4-odd-1-even instead of 5-even/5-odd will give you 6 times more ways to win. You also have 322,920 fewer ways to fail.

We are looking for the best ratio, so the search is not over yet.

3. 2-odd-3-even/3-odd-2-even

Either pattern offers 845,000 ways to win and 1,753,960 ways to fail. If you play with the 2-odd-3-even/3-odd-2-even pattern instead of a 1-odd-4-even/4-odd-1-even pattern, you can have 2 times more ways to win. This will also reduce your ways to fail by 456,300.

Compared to 5-even/5-odd, 2-odd-3-even/3-odd-2-even gives 13 times more ways to win. You could also have 779,220 fewer ways to lose. Thus, the most favorable combinations are those with a2-odd-3-even/3-odd-2-even pattern.

The odd-even composition is just one part of the basic combinatorial analysis. To complete this process, you should also look at the low-high composition of combinations.

# Which Combinations Have The Best Low-High Ratio?

A 5/52 lottery game has the low and high number sets of

Low = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26}

High = {27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52}

The low-high patterns for this game are:

1. 5-high/5-low

Either pattern has a probability value of 0.02531012404962. Therefore, in 100 draws, you have 3 opportunities to match the winning combination.

Each pattern has a ratio of success to failure of 1 to 39. This could mean taking part in 40 draws to get your one chance at winning.

2. 1-low-4-high/4-low-1-high

The 1-low-4-high or 4-low-1-high pattern has the probability value of 0.14955982392957. Thus, there are 15 opportunities to win in 100 draws. This is 5 times more than the estimated occurrences of 5-high/5-low.

With the ratio of 1 to 6, either pattern might require 7 attempts before you get your best shot at winning. The 1-low-4-high/4-low-1-high has 7 times better ratio of success to failure than 5-high/5-low.

However, 1-low-4-high/4-low-1-high is still not the best choice.

3. 2-low-3-high/3-high-2-low

The probability value of each pattern is 0.32513005202081. Hence, the estimated occurrences in 100 draws are 33. This is 18 times more than 1-low-4-high/4-low-1-high and 11 times more than 5-high/5-low.

2-low-3-high and 3-high-2-low each offers a ratio of 1 to 2. Three attempts could give you the opportunity to tally with the winning combination. This is the most sensible choice for you, since it requires the least number of attempts.

You also spend the least amount of money before having your shot at winning. 2-low-3-high and 3-high-2-low offers 3 times better ratio than 1-low-4-high/4-low-1-high and 20 times better ratio than 5-high/5-low.

Every combination shares an equal probability with each other.

Choosing a 3-low-2-high combination over a 5-high combination does not give you a higher probability of winning. All combinations in a lottery game share an equal probability. However, a 3-low-2-high combination is a better choice than 5-high when you consider their ratios of success to failure. A 5-high combination has more ways to make you fail, but fewer ways to help you win.

There are two separate analyses for odd-even and low-high compositions. Ultimately, however, you must consider both analyses in determining if a combination is worthy to play.

Let me show you how to apply the conclusions of the two analyses by considering the samples combinations above.

1. 1–2–3–4–5

This combination has the best odd-even ratio (2-odd-3-even), but the worst low-high ratio (5-low). Hence, a perceptive player will not continue using this. Instead, he will change some numbers to improve the ratio.

2. 48–49–50–51–52

This is also a combination with best odd-even ratio (3-odd-2-even), but worst low-high ratio (5-high). It also requires changing some numbers for achieving a better ratio.

3. 10–15–16–22–52

This is an example of a combination with good/average odd-even (1-odd-4-even) and low-high (4-low-1-high) ratios. It also means you need to change some numbers to achieve better results from your games.

Again, these are only sample combinations. I do not recommend that you use these when you actually play a 5/52 game.

Changing some numbers is crucial since the odd-even ratio must agree with the low-high ratio (vice versa). Proceeding using a combination with contradicting ratios would not be a productive step.

Apparently, when the need to change some numbers arises, repeat the basic combinatorial analysis. The new combination may have a different ratio of success to failure than the first combination you analyzed.

This can be tiring and perplexing, especially for someone who has no passion for math. Thus, you could instead use advanced combinatorial analysis.

# Advanced Combinatorial Analysis For Finding Best Patterns And Combinations

To play the lottery successfully, you need to follow a mathematical game plan. (To find out more, be sure to read How to Win the Lottery According to Math)

Math can offer many solutions to a problem. Some solutions involve more steps than the others. The more steps involved, the more mistakes you must avoid. Advanced combinatorial analysis is more similar to the solution with fewer steps, but its results remain accurate and precise.

It mainly eliminates the possible contradictions of ratios. It provides a convenient way of applying combinatorics and probability to your lottery games.

In this table, the odd, even, low and high numbers have been regrouped. Now, you can easily tell which numbers are low-odd, low-even, high-odd and high-even.

You no longer need to write 4 separate sets of numbers, which could often result in errors. Using an advanced combinatorial design, you could quickly discern if a combination is good to use in the game. Let us use this for our examples above.

1. 1–2–3–4–5

The realization could hit you almost upon looking at the advanced combinatorial design. Practically instantly, you know this is not a combination to even think about spending money on.

2. 48–49–50–51–52

Looking at an advanced combinatorial design, you will know without even pressing a button that this is one of the least favorable combinations in a 5/52 game.

3. 10–15–16–22–52

The advanced combinatorial design will also reveal that this is a combination not worthy of your time and resources.

Combinatorial analysis, both basic and advanced, implements balance in composition. Balance can critically determine the best ratio of success to failure. Through advanced combinatorial analysis, however, you may not deal with complex computations.

Still, you’d be able to know the best, good, bad and worst combinations. This is what combinatorial analysis is about. It lets you know your options, but leaves the decision making onto your hands.

Do math computations overwhelm you? Then use a calculator to make the process easier. (See Lottery Calculator: Knowing the Best Lotto Combinations Without a Math Degree)

As a lottery player, it is important that you know your options. However, it is more important to not waste your ability to choose the best. You can play using any combination, even though it is the worst choice. Still, logic dictates that you choose combinations and patterns with best ratios.

Allow me to illustrate why.

A 5/52 lottery game has 56 advanced combinatorial patterns. Out of 56, there are only 4 best patterns. Many players, such as those relying solely on past draw results, might be naively spending money on combinations with patterns #5–56.

They play non-stop; inspired by their misconceptions that they will one day win. They might be playing and waiting all their lives, but have yet to win anything significant.

Through advanced combinatorial analysis, you can pursue the jackpot with a definite direction from the best patterns. Playing with the best patterns and combinations is not exactly a shortcut. Instead, it ensures that you will reach your destination with fewer detours.

Playing with best patterns and combinations help ensure you are wisely spending money, time and resources. This is what responsible lottery playing is about. (Read Play The Lottery Responsibly to know more.)

It is understandable if some people remain skeptical about preferring the best combinatorial patterns. After all, there are still some people who luckily won the lottery jackpots, making no computations or analysis.

Still, let me show you the worth of playing with the best patterns.

1. Best Pattern

Expected Occurrences (Pattern #1)

= 0.0659363745 x 2,000

= roughly 132 occurrences

2. Middle Pattern

Expected Occurrences (Pattern #5)

= 0.0304321729 x 2,000

3. Worst Pattern

Expected Occurrences (Pattern #29)

= 0.0085834334 x 2,000

In 2,000 draws, pattern #1 may occur 132 times. It has 330 expected occurrences in 5,000 draws.

In 2,000 draws, pattern #5 may have 61 occurrences. Expect that it can occur 152 times in 5,000 draws.

In 2,000 draws, pattern #29 may occur 17 times. It has 43 expected occurrences in 5,000 draws.

The best patterns will dominate the game. The law of large numbers will be fulfilled at infinity. So, an increase in the number of draws will show the same trend.

In 2,000 draws, pattern #1 offers 2 times more opportunities to win than pattern #5, a middle pattern.

Pattern #1 also offers 8 times more winning opportunities than the worst pattern, pattern #29.

Winning opportunities refer to the chance of matching the winning combination in a specific number of draws. For example, you want to win at least the second division prize in Lotto America, using the best pattern is a productive idea.

Lotto America is a 5/52 game with an extra ball from 1 to 10, so the total combinations become 25,989,600. This makes winning the jackpot (5-number combination + Star Ball) extremely hard. Still, winning the second prize of \$20,000 when you match the 5-number combination is already a huge success.

It is here where advanced combinatorial analysis can be of big help. Playing with the best pattern can take you closer to the prize. There are two draws per week for Lotto America.

Thus, 2,000 draws might take about 20 years, which is playing for almost a lifetime. Let us reduce our time frame to a year with about 104 draws. Suppose you have allotted \$104 so you could play all the 104 Lotto America in a year.

Which pattern do you think will provide the best value for money?

1. Best Pattern

Expected Occurrences (Pattern #1)

= 0.0659363745 x 104

= roughly 7 occurrences

2. Middle Pattern

Expected Occurrences (Pattern #5)

= 0.0304321729 x 104

3. Worst Pattern

Expected Occurrences (Pattern #29)

= 0.0085834334 x 104

The only patterns that could occur in 104 draws are pattern # 1 and #5. Therefore, a worst pattern could not give a good game even once in a year.

Between pattern #1 and pattern #5, the better choice is pattern #1. It offers 4 more opportunities to win than pattern #5. Hence, the best patterns like pattern #1 can give the best value for your money and effort.

In terms of the number of attempts you must make for an opportunity to win, the pattern #1 remains the most favorable.

Pattern #29 offers a ratio of 1 to 118. You might need to play 119 times and spend at least \$119 before you have the chance to win. This could mean playing twice a week for 15 months.

With a ratio of 1 to 33, pattern #5 will give you this opportunity after about 34 attempts or draws. As a middle pattern, it can improve your game than when you use a worst pattern. You probably need to play for 4 months to get a good opportunity. Still, this is just a middle pattern.

Pattern # 1 provides a ratio of 1 to 15. This is 2 times better than pattern #5. You would need to spend at least \$16 for 16 draws in 2 months for a chance to match the drawn combination.

This shows that the best pattern can surely make your lottery budget well spent. Apart from helping spending your money wisely, combinatorial analysis also suggests a timing strategy.

Remember, patterns occur at an interval. The table above shows an approximate interval. Pattern #1, for example, has 15. This is only an estimated interval because it is impossible to know the exact value and occurrences.

Still, if pattern #1 occurred today, it is improbable to occur in the next draw, but after 13–17 draws. You may skip playing when you do not expect the pattern to occur. Save money at this idle period so you can buy more lines on your next games.

Prefer a combination offering the best ratio of success to failure.

The probability of lotteries is not something anyone can change. The odds of winning are also not that easy to outplay. This does not mean you should be hopeless. As a lottery player, you could know what your options are. You can choose smartly, so choose the best ratios of success to failure. Use the ratios of success to failure to decide the best course of action. You may even choose not to play, if you think it is the best action to make.

# Should You Still Play Lotteries Using Past Draw Results?

Mathematics offers the best way to play the lottery. Still, every strategy that can help you win is welcome. If you decide to still use past draw results when playing lotteries, you are free to do so. Yet, let me remind you never spend money on these combinations unless you have determined they have the best ratio of success to failure.

Edvin Hiltner runs Lotterycodex.com to prove that a sensible lotto strategy is practically feasible through the use of mathematics.

## More from Edvin Hiltner

Edvin Hiltner runs Lotterycodex.com to prove that a sensible lotto strategy is practically feasible through the use of mathematics.