A Truly Random Lottery with a Deterministic Outcome

Last updated on February 22, 2024

It’s hard to win in a truly random lottery. Initially, its randomness might be a stumbling block, but when you look at it deeply, it might be a stepping stone for you as a lotto player. You might as well be thankful that the lottery is truly random.

An idea appeared on top of my head. If there’s a picture that describes the randomness of the lottery, what would it possibly look like?

Quickly, I created a computer simulation program. One way to get my objective done is to imitate the random lottery draw process and paint an image out of it.

And this was the picture I got:

Initially, it may not be obvious, but the picture suggests specific ideas on not being mathematically wrong when playing the lottery. It indicates that a lottery game is mathematically deterministic.

We will delve into this mathematical strategy later. But in the meantime, let me give you a hint:

Mind you, with this mathematical strategy, you should be thankful that you’re playing a truly random lottery game. If the lottery is not completely random, then any probability calculation cannot be right.

Science knows only two types of processes. One is deterministic, and the other is random. If you combine the two, you get something probabilistic.

Most of us fail to see how the lottery works because most lotto players still think that statistics is the right tool to analyze the game. For example, people look for hot and cold numbers.

I think it’s high time to correct this mistaken notion because you don’t use the tools of statistics when finite numbers are involved. Since the lottery has finite possibilities, any questions we ask about its possible outcome are always combinatorial and a probability problem to solve.

Does that mean we can predict the lottery?

If you mean the next winning numbers, then, of course, it’s not possible. But if we try to predict the general behavior of a truly random lottery game from the perspective of the law of large numbers, then yes. It’s mathematically possible.

In a random event like the lottery with finite possibilities, the actual results always agree with the probability calculations.

Therefore, the lottery is deterministic because a random draw obeys the probability dictate. This conclusion can be useful in many ways (for example, how to be wrong less for most draws).

That’s how mathematical information matters. When you know all the possibilities, you have the power to make informed choices, and you will never be mathematically wrong. One example is the calculation of success-to-failure ratio. See my post: The Winning Lottery Formula Using Math.

So, complete randomness is a requirement to be able to make mathematical predictions.

But don’t get me wrong. Even 1-2-3-4-5-6 is as likely as any other combination. This article is not about probability per se.

That’s the problem when we are frequently exposed to probability and completely forget the concept of odds. Odds and probability are not mathematically equivalent. We will talk about the big difference in detail below.

Again, I would like to reinforce that a true mathematical strategy is choosing the best ratio of success to failure.

In this article, I will not discuss how to win the lottery. If you view this article that way, then you’re welcome. Please read my article: How to Win the Lottery According to Math.

With all the noise on lottery strategies online, I say mathematics remains the only strategy that works.

But suppose you are not interested in the nifty method of calculations involved. In that case, you’re lucky because you can use Lotterycodex calculators to utilize all these combinatorial and probability principles and understand your game better.

This article does not intend to create an illusion of control.¹ You cannot control the outcome of any lottery draw. The lottery is gambling. What I mean to do here is to show you pieces of mathematical information that will get you to look at the lottery in a new light.

Let’s return to the picture of the lottery’s randomness (shown above). First, notice the evidence of streaks and clusters. Why does this clustering take place?

Well, clustering is an intrinsic characteristic of random data. But that’s not the important question. The most important thing is the message.

You should ask, “What does it tell me?“

There are more stories to the picture than meets the eye.

So stay tuned because this article will explain why a truly random lottery is mathematically predictable.

It’s going to be a long article. So be ready.

Let’s begin.

Simulating a Truly Random Lottery Game

Here’s the question that comes to mind. How can I paint a picture of a lottery to describe its randomness?

I have seen the work of Steven Pinker in his book The Better Angels of Our Nature. And I’ve seen this visual comparison by Bo Allen.²

But I want something that reflects the nature of a random lottery and also pick a strategy that I can explain.

It’s not an easy task. But I have a clear goal: the only way forward is to start somewhere.

First, I think of and choose a minimal lotto game format. A 4/20 lottery game can only produce a total of 4,845 playable combinations. So, I thought this format was quite manageable.

Fortunately, while the lottery rules may differ from operator to operator, the draw mechanics are the same.

Pick one ball at a time while the balls get shuffled inside the drum. With each pick, the balls in the drum get decreased by one. Drawing continues in this manner until the winning combination is complete.

So, if I have to create a simulation program, it must do the following:

Step 1: The program will shuffle the set while the system picks one number.

Step 2: The number will be removed from the set.

Step 3: Back to step 1 and repeat the process until the winning combination is completely drawn.

The program saves the results of the drawing in a database. It’s crucial to run the process more than a thousand times to obtain a large enough data set for analysis.

This large data set will help provide conclusive evidence from the perspective of the law of large numbers. From this large data set, the program will count the observed frequency for each combination drawn many times and draw out the information visually in a particular way.

The first square will represent the 1-2-3-4 combination. 17-18-19-20 will occupy the last square.

Each time a combination is drawn during the simulation test, the gray represents one occurrence, and then the shade gets darker as the occurrence increases until it turns red. Red color means that a combination was drawn more than ten times, and white means that it was never drawn and had zero frequency.

It is a seemingly simple idea, but it’s much easier said than done. First, I need to ensure a couple of things are done correctly. One of the issues is the quality of the random process that the simulator will use.

Random is Random. Well, Not Exactly

When we allow a computer to generate random numbers, we use a pseudo-random number generator, or PRNG for short. How can a precise machine generate numbers by chance? It is difficult for a computer to generate random numbers because it follows a set of instructions blindly, and any output it produces must be predictable.³,⁴

But random processes are everywhere; therefore, computer scientists must embrace it.

So, I use several practical implementations to create randomness draw after draw. However, some of these methods are unsuitable when predictability is critical.

Have a look at the following PHP5 snippet:

The script above will produce the following numbers:

Can you see a pattern?

One can easily know the next numbers on the list. All it takes is to identify the initial seed used to generate the numbers.⁵

PRNGs can produce numbers that appear random, but in reality, they are predictable. If I run the same script on a different machine, I will always get the same results.⁶

Let me show you visually how deterministic PRNGs are in a simulation program.

I must show you what it looks like because you must see the difference between a nonrandom and a genuinely random picture.

Using the 4/20 lotto format, here’s how predictable a computer simulator can be:

Can you see the clustering pattern doesn’t change? Each time the computer runs, the simulator draws the same combinations.

At 1000 draws, the simulation program distributes the frequency in 896 combinations. However, if run at 3000 draws, the program seems to pick numbers from the same combinations.

As a result, the clustering pattern is constant. Try to compare the corners.

If a lottery game operates like this, you immediately know there’s a loophole. All you have to do is pick your combinations from those 896 combinations and forget the rest. Therefore, winning is just a matter of time.

Lottery operators use a true random number generator program. But how do we know? It’s always best to ask. And for you lotto players, you have the right to know.

Canada moved Lotto 6/49 and Lotto Max to computerized drawing on May 14, 2019.⁷ I wrote an article addressing the people of Canada to ask ALC how this computer-based drawing can protect the integrity of the draw.

At the time of writing, the only information in the press was a statement from an official. He only assures the integrity of the draw with the presence of two sets of auditors for check and balance. There’s no statement about how the computer generates random numbers.

In the United States, I am aware of lottery systems that use computerized drawings, including the one successfully rigged by Eddie Tipton (now in prison).⁸

Most do not know how numbers are generated randomly from the computer code level.

Going back to random generators, as you can see, a lottery simulation program requires a truly random process. And carelessness is the last thing to do if I should seek an accurate picture of a truly random lottery.

The Hunt for a True Random Number Generator (TRNG)

One must consider an unbias and unpredictable process in creating a lottery simulation program. This objective is especially true for the lottery and all types of gambling.

One method is introducing external random data to the computer system to seed and reseed the PRNG.

A simple method that comes to mind is counting the hits one of my websites received. The points in time at which people visit and click on a website are completely random and non-deterministic and, therefore, can be a good source of random information with a certain degree of entropy. This method, though, requires a bit more coding work on my end.⁹

Computer technology has advanced so much that you have several good options available.

One method is to use a physical phenomenon. For example, you can measure radioactive decay using a Geiger counter connected to a computer. This method is not a practical solution for my small simulation project.¹⁰

Another option is the use of the Random.org service, which is free. The Random website generates randomness from atmospheric noise. Although free, I have to set up a system to request data from their server and ensure I don’t exceed the quota. I can quickly reach the limit for a lottery simulation program in no time.

I’m thinking of something I can use immediately for my random test generator machine.

Fortunately, nowadays, programming languages have evolved to address the issue of randomness.

PHP7 introduced a cryptographically secure pseudo-random number generator or CSPRNG with properties that make number generation unpredictable. ¹¹,¹²,¹³

This CSPRNG can be easily invoked with the use of random_int directive.¹⁴

It’s good to know we have a cheaper and readily available solution.

But a series of statistical tests must be done to make sure that random generation is of high quality.¹⁵

A Test of Quality Randomness

Since a practical option is available using PHP7, my next step is to check if CSPRNG does what it promises.

The first test is to check if the random distribution that the code produces abides by the law of large numbers.

In the lottery, all numbers have an equal probability. According to the law of large numbers, all numbers converge around the same expected value if an experiment is repeated many times. Therefore, a simulation program must be able to produce the same characteristics.

If we have a set of 20 numbers and pick one at a time, each number will have a probability of 1/20 or an expected value of 5 in 100 draws. If we pick one million times, each number should have an observed frequency of more or less 50,000.

The above table looks good. But we can have a better look at it visually using the pie graphs below:

As you can see, each number gets the same share of the pie, which indicates unbiased selection. Random_int passed the first test.

Next, we will check the quality by which numbers are generated randomly. We need to check how close the results are to the expected value.¹⁶

The best way to check is to compare random_int with a non-CSPRNG counterpart called mt_rand in a popular dice roll experiment.¹⁷

A dice has six numbers. And each number has a probability of 1/6. If we throw three dice at the same time one million times, the computer program must be able to reproduce the following expected values very closely:

I have to compare two PHP7 functions against each other with a series of statistical results.

Here are the initial results from the first run:

As you see, the two functions are closely matching the expected value. The table indicates that random_int is doing better. However, many repeat tests are essential for comparison.

Below is a full list of my sample tests:

We can visually see how the two random functions fare against each other using the graph below:

To know which produces random numbers better, we must look for the one plotting closer to the zero line. From the graph, the CSPRNG’S random_int wins the quality test.

Creating the Simulation Program

My initial tests show I can proceed with my simulation of a truly random lottery game using the random_int directive.

I used the 4/20 lottery format.

You will expect a lottery simulation program that doesn’t produce bias and a predictable clustering pattern.

That means even those unusual combinations such as 1-2-3-4, 17-18-19-20, or 5-10-15-20 need to occur.

A truly random lottery draw must allow all combinations to occur if we give the lottery enough opportunities. According to the law of truly large numbers, even the most unusual combinations, coincidences, and rare events must occur.

In other words, seeing a streak of unlikely combinations in a row is unsurprising.

So let me show you now the results of my simulation program:

Let’s continue observing the behavior of the draw up to 5000 draws:

At 5000 draws, you cannot expect that all the 4,845 combinations will get drawn. It will take around 40,000 to 50,000 draws before all the 4,845 combinations are drawn.

Let’s continue to 15,000 draws.

At 15,000 draws, some combinations exhibit more frequency than others. And many combinations still haven’t been drawn yet, as demonstrated by the existence of white spaces.

You might ask, why are there 227 combinations that have yet to occur. The probability theory has an answer. We will talk about this probability principle later.

Notice that at this stage, you are beginning to see red squares. Those are combinations that have been drawn more than ten times.

Fast forward to 45,000 draws; this is what the simulation image looks like:

You will notice that at this stage, all combinations have been drawn. As I said earlier, a random lottery must follow the law of truly large numbers.

A real random lottery draw must allow all combinations to occur if we give the lottery enough opportunities. According to the law of truly large numbers, even the most unusual combinations, coincidences, and rare events must occur. This means that even 1-2-3-4 or 2-4-6-8 should occur.

Interpreting the Lottery’s Random Behavior, Probability, and the Odds

Combinations are not created equally. Allow me to explain why I said it.

There’s only one way to win the jackpot prize.

So, no matter what we do, we can’t change the probability.

For example, in a 6/49 game, 1-2-3-4-5-6 is only one of 14 million combinations.

All of the distinct combinations in a 6/49 lottery game possess the same probability.

However, if you keep looking at the probability of an individual combination, you only see one small portion, not the whole picture of the lottery.

You must investigate the game and analyze how everything works from a bigger picture.

Deep within the finite structure of the lottery are layers of combinatorial compositions that may provide a better clue and make us look at the lottery in a new light.

In statistical science, probability and odds are two different terms and have two different equations.¹⁸

The likelihood of winning a truly random lottery is expressed in probability percentage, but what matters to you is the odds or the ratio of success to failure.

We can explain how this success-to-failure ratio works using combinatorial composition.

Remember the dice roll?

Those dice with total sums of 10 or 11 will dominate most of the rolls. No matter how often we repeat the experiment, the outcome will always be the same and deterministic.

The smallest sum that a three-dice roll can produce is 3.

1 + 1 + 1 = 3

Then, the highest sum possible is 18. That is:

6 + 6 + 6 = 18

There’s only one possible way that three dice can total 3. The same holds for the sum of 18.

Therefore, the probability is 0.5% for each. A probability value like that can only occur about five times in 1000 draws.

A three-dice combination with a total of 11 has a probability value of 12.5%, and therefore all three-dice that total 11 are expected to occur about 125 times in 1000 draws.

The lottery works the same way.

In a 4/20 lotto game, the smallest sum is 10.

There’s only one way to combine numbers that total 10.

1+2+3+4 = 10

Therefore, this combination has a probability value of 0.00020639834. Simply put, a 1-2-3-4 combination may only occur twice in around 10,000 draws.

In other words, when you play 1-2-3-4, you have the advantage of a sure win since it’s the only one in its group. However, the success-to-failure ratio indicates you get one favorable shot after about 5000 attempts.

On the other hand, some combinations have a total sum of 44. This group has a probability value of 0.03591331269. This group occurs about 359 times in 10,000 draws.

Since there are only 174 combinations that total 44, then you have 174 opportunities to win against 4671 times that you don’t. The success-to-failure ratio indicates that you get one favorable shot every 27 attempts.

Sum of 10 VS Sum of 44

As you can see, the success-to-failure ratio is an important gauge for lottery players.¹⁹

Combinations have different compositions. We put combinations that share the same compositions into combinatorial groups.

These combinatorial groups have varying success-to-failure ratios. These combinations with the best success-to-failure ratio are the dominant compositions in lottery draws.

Making the right choice is easy when you know all the possibilities.

You don’t want to spend your money with a group that only gives you one favorable shot after 5000 attempts. Do you?

Your goal as a lottery player is not to manipulate a random game. Rather, you aim to play your best shot by playing the dominant composition. It is the success-to-failure ratio that will tell you which one is.

The Wrong Use of Statistics

If you want to know the dominant combinatorial template in your lotto game, then statistics is not the appropriate mathematical tool.

Statistics often fail because it tricks you into believing something works until enough data proves it wrong.

Probability and statistics are two distinct concepts that approach a problem differently.

And the main difference has to do with what we know.

So, depending on our knowledge, a problem could be either statistical or probabilistic.

For example, we have a box of 20 marbles. We know there are yellow, cyan, gray, and green marbles inside the box, but we don’t know how many each color has.

Since we do not know the composition of the 20 marbles, we use the tools of statistics to infer the composition of the marbles based on a random sample.

If, instead, we know that there are five yellow, five cyan, five gray, and five green marbles, then any question we have is an applied probability problem to solve.

So, with the use of probability, we can ask questions such as:

What’s the likelihood that we draw one yellow, two cyan, and one gray marble?

or

What’s the probability that we draw all four green marbles?

In other words, we don’t need statistical analysis when our knowledge is adequate to answer a problem.

The same concept works in the lottery.

We can ask:

What is the probability that we draw 1-2-3-4?

This question is simply a matter of rephrasing the question into:

What is the probability of drawing 2-low-odd and 2-low-even numbers?

And we can obtain the answer without resorting to random samples or statistical analysis.

That’s how probability works in the lottery. And it works in all kinds of lottery systems, whether 5/50, 6/49, 6/42, or 5/35.

Probability gives you the powerful knowledge to guide and help you get the best shot possible.

Surprisingly, despite the discovery of probability theory earlier in 1700 by the famous French mathematicians Blaise Pascal and Pierre de Fermat, using statistics to pick the best numbers in the lottery has never been corrected.

However, it is interesting to note that more than a century earlier, an Italian polymath, mathematician, and accomplished gambler, Girolamo Cardano, had figured out the mathematical solution to probability problems long before Pascal and Fermat did.

It’s high time you change the way you look at the lottery.

Of course, probability analysis in the lottery is impossible without the help of another branch of mathematics called combinatorics.

So here in Lotterycodex, we combine combinatorial mathematics and probability theory to determine the dominant combinations in the lottery.

PROBABILITY and COMBINATORICS = PRECISE AND ACCURATE LOTTO PREDICTION

The results of Lotterycodex calculations are high-precision and high-accuracy prediction, which statistics fail to provide.

Of course, I am not talking about predicting the next winning numbers. I am talking about the whole picture of the lottery from the point of view of the law of large numbers. You can see the future of your game, as you will discover later.

With enough opportunities, probability calculation is always a mathematical certainty.

Let’s discuss now how combinatorial math and probability theory work together as a lotto strategy.

Lotterycodex Combinatorial Analysis

What do you feel about the people who play 1-2-3-4-5-6?

Or those that pick numbers like 5-10-15-20-25-30?

I guess you think those players won’t likely hit the lottery, and if they did, they would probably share the prize with many players.

But there’s more to that than meets the eye.

A truly random lottery doesn’t like drawing numbers that follow regular patterns.

You see, people may avoid 10-20-30-40 or 2-4-6-8.

But these combinations are easier to detect. Some combinations have regular patterns that are less noticeable.

If you’ve been playing the lottery, chances are, you have played one of these combinations many times and aren’t even aware of it.

Earlier, we discussed the three dice roll, and you can group combinations based on their total sums.

However, the sum doesn’t provide a clear-cut rule on what makes up a combination with a better success-to-failure ratio.

For example, a lotto player may pick the combination 01-07-17-19 because he believes it falls within the best sum range.

Of course, it falls within the best sum range.

However, 01-07-17-19, composed of pure odd numbers, cannot have the best success-to-failure ratio.

Just because a combination falls within the best sum range doesn’t mean it’s a good bet. The sum range doesn’t provide a granular clue on what numbers to pick. That’s why I don’t recommend using sum as a strategy.

So, instead of adding numbers, Lotterycodex proposes a unique combinatorial design that can handle low-high and odd-even numbers in one probability calculation.

To illustrate, Lotterycodex divides the 4/20 game into four sets:

The above sets will serve as a detailed guide on picking numbers by simply following a template.

An example of a template could be a composition of 2-low-odd, 1-high-odd, and 1-high-even numbers. This combinatorial template can be presented visually using colors. So, a list of combinations that follow this template are the following:

These combinations share the same probability of 0.0515995872. This group only occurs approximately five times in 100 draws. In Lotterycodex, these combinations belong to template #11.

Based on the Lotterycodex combinatorial calculations, a 4/20 game has 35 templates. The dominant one is template #1.

I will show you next to compare one template against the other to prove that the lottery is deterministic to an extent.

In this context, I am not talking about determining the exact winning combination. Instead, I’m discussing predicting the exact combinatorial template dominating the winning draws. According to the law of large numbers, the same template will continue to dominate as the number of draws gets larger and larger.

Template #1 Dominates the 4/20 Draws

According to the law of large numbers, we can expect that template #1 will dominate the lottery draws and continue to dominate the 4/20 game as the drawings grow.

Template #1 has a probability value of 0.1289989680.

To predict the expected frequency of this template, multiply the probability by the number of draws.

Expected frequency (template #1) = 0.1289989680 x expected number of draws

For example, in 100 draws, we expect template #1 to occur approximately 13 times.

Expected frequency (template #1) = 0.1289989680 x 100 = 13

In 1000 draws, we expect this template to occur 129 times.

Expected frequency (template #1) = 0.1289989680 x 1000 = 129

Then in 5000 draws, template #1 will occur 645 times.

Expected frequency (template #1) = 0.1289989680 x 5000 = 645

Do the same calculations for the other templates, and you’ll see that template #1 dominates at any time.

To illustrate, let’s compare template #1 against template #2 and show the difference visually.

Template #1 VS Template #2

In the following visual comparisons below, the red color represents template #1. Blue represents template #2. In all the images, notice the dominance of template #1.

100 draws

Template #1 is expected to occur 13 times, while template #2 is expected to occur five times.

Expected frequency (template #1) = 0.1289989680 x 100 draws = 13

Expected frequency (template #2) = 0.0515995872 x 100 draws = 5

The actual draws show that the probability estimation is very close. We cannot expect the prediction to match since probability theory is only a mathematical guide. But is it a reliable tool to predict the likely outcome of the two templates? You bet.

500 draws

In 500 draws, template #1 should occur approximately 64 times. Template #2 should occur about 26 times.

Expected frequency (template #1) = 0.1289989680 x 500 draws = 64.49

Expected frequency (template #2) = 0.0515995872 x 500 draws = 25.79

See, the actual draws are pretty close. Let’s go forward to 1000 draws.

1,000 draws

Expected frequency (template #1) = 0.1289989680 x 1000 draws = 128.99

Expected frequency (template #2) = 0.0515995872 x 1000 draws = 51.59

As usual, the actual draws always agree with the probability calculation. According to the law of large numbers, template #1 must dominate the results of the 4/20 lotto game. It’s a mathematical certainty that the lottery is subordinate to the dictate of probability theory. The study of templates is so powerful that John Francis proposes to put all these draw game mysteries in a specialized field of study called draw pattern mathematics or DPM.

Look at a visual comparison as the drawings get larger and larger, from 3000 to 5000 draws.

3,000 draws

Expected frequency (template #1) = 0.1289989680 x 3000 draws = 386.996904

Expected frequency (template #2) = 0.0515995872 x 3000 draws = 154.7987616

5,000 draws

Expected frequency (template #1) = 0.1289989680 x 5000 draws = 644.99484

Expected frequency (template #2) = 0.0515995872 x 5000 draws = 257.997936

The Mathematical Certainty of Lotterycodex Analysis

For a 4/20 game, Lotterycodex recommends players choose the dominant template #1.

The visual comparisons below will prove that:

Template #1 VS Template #5 in 5000 Draws (Template #1 wins)

Template #1 VS Template #10 in 5000 Draws (Template #1 wins)

Template #1 VS Template #15 in 5000 Draws (Template #1 wins)

Template #1 VS Template #20 in 5000 Draws (Template #1 wins)

Template #1 Vs Template #25 in 5000 Draws (Template #1 wins)

Template #1 VS Template #30 in 5000 Draws (Template #1 wins)

Template #1 VS Template #35 in 5000 Draws (Template #1 wins)

The above graphs show that you should not worry about all the other templates. You can be sure to get the best shot possible if you stay with template #1.

The lottery may be truly random, but sticking with template #1 is based on the fact that the lottery is mathematically deterministic to an extent.

Something to Try When Playing a Truly Random Lottery

As a lotto player, you play to win. Your job is to get the best shot possible.

Please read my other articles:

Lottery Wheel: A Clever Mathematical Strategy That Works

How to Win the Lottery According to Math

The Winning Lottery Formula Using Math

You need a system that will provide a granular guide on what numbers to pick. That’s how combinatorics and probability theory will come in very handy. However, the calculation can be too exhaustive and tedious in a random lottery with many numbers. That’s why Lotterycodex offers a lottery calculator to do the heavy lifting.

For example, here’s what a Lotterycodex calculator will show you for a 6/49 lottery game:

Generated by Lotterycodex Calculator

Based on the table above, you should focus your strategy on that dominant group to get the best shot possible. For the Lotto 6/49 game, templates #1, #2, and #3 will dominate the game.

Here’s another example of a Lotterycodex analysis for a 5/69 lottery game such as the Powerball:

Generated by Lotterycodex Calculator

If you’re a Powerball player, you should focus your strategy on template #1.

When using a Lotterycodex calculator, please ensure you use the right format. Lotterycodex has the right analysis for all kinds of lottery games worldwide.

Thank you for reading. Please remember that a lottery game is not an investment. Play the lottery for fun.

Questions and Answers

Additional Resources

1. Illusion of Control    [↩]
2. Random.org: Simple Visual Analysis    [↩]
3. Introduction to Randomness and Random Numbers    [↩]
4. Pseudo-Random Number Generator (PRNG)     [↩]
5. Not So Random: Exploiting Unsafe Random Number Generator Use    [↩]
6. Insufficient Entropy For Random Values    [↩]
7. The Great Lottery Debate: Computerized Draws vs. Lottery Balls    [↩]
8. The untold story of how a gaming geek with a checkered past pulled off the biggest lottery scam in U.S. history    [↩]
9. Entropy (information theory)     [↩]
10. How can a totally logical machine like a computer generate a random number?    [↩]
11. Cryptographically Secure Pseudo-Random Number Generator (CSPRNG)
        [↩]
12. Wikipedia: Cryptographically secure pseudorandom number generator    [↩]
13. Secure Randomness in PHP    [↩]
14. PHP random_int    [↩]
15. Randomness_test    [↩]
16. Randomness in PHP – Do You Feel Lucky?    [↩]
17. PHP: mt_rand    [↩]
18. Odds, Probability, and the Lottery    [↩]
19. The Lotto Secret: Three Math Strategies for Winning Revealed    [↩]