How to Hack the Lottery
by StankDawg
So you want to win the lottery...
Overview
Most states have a lottery these days. Even though gambling is illegal in most states, somehow the lottery is different. I won't go into explaining the hypocrisy in that scenario, as that is not the point of this article. It should suffice to say that the money is supposed to go to the state governments, which justifies the exclusion from the rules.
Regardless of that debate, I would like to shed some light on how the lottery works and settle the debate on why (or why not) to play the lottery. I will use some formulas and mathematical functions to explain the logic, but hopefully the text of this article will teach you how to analyze your specific lottery and not rely on the specific examples that I used. I think the point will still be understood.
Logistics
Let's talk about how the lottery works. First of all, it is important to know that each state's rules may vary, but they usually have some physical procedures in common. Most states use different sets of ping-pong balls that they rotate in and out of use. This is to avoid the possibility that a set may have something wrong with it which could skew the odds. It could have a ball that is lighter than the others, has a hole in it, or that could be dirty. Along the same lines, the machines that pick the balls are usually rotated in and out of use and calibrated regularly as well. This prevents the machines from malfunctioning and ensures that they haven't been tampered with. Finally, to make sure that the controlled environment stays controlled, an independent auditing firm verifies all of the equipment, the environment, and the people involved are all checked to avoid foul play. The bottom line is that this is a controlled environment! You have to accept that to continue.
Each state varies, but let's pick some arbitrary examples. Let's say you have to match six numbers, in any order, out of balls numbered 1 through 50. You pick six numbers hoping to match all six of the balls pulled from the tumbler. When the first ball is pulled, you have a 6 in 50 chance of being correct with one of your numbers. That is pretty clear common sense thinking right? O.K., so you actually get lucky and one of the numbers you had is pulled from the tumbler! Lucky you! Now on to ball two.
So, the first ball is drawn and now there are 49 balls left. You still have five numbers to match. Your chances of getting the next pick are even better now that there are only 49 balls left, right? Not exactly... as a matter of fact, not even close.
Statistics
Let's preface by saying that all numbers are rounded for the sake of readability. Now the specific area of statistics we are discussing here is probability. What are the chances that an event will happen? You have given information to begin with and a mathematical basis upon which to calculate. The most helpful concept is that of a factorial.
A factorial is notated using a "!" after the number. It usually is located on your scientific calculator as "n!". 3! is a factorial of 3, which simply means (3 * 2 * 1) which is 6. That one is easy to do in your head, but what is 50! without using a calculator?
Now don't go get all bent out of shape. It is a long process with lots of numbers, but it isn't as difficult as it sounds. You can calculate the probability of each individual pick and then multiply them all together to get the final probability. Note that the order of the numbers is unimportant. It doesn't matter if your picks are in the same order as the drawing. If they were, it changes everything and the odds skyrocket astronomically.
Luckily, there are formulas that we can use to apply the factorial notation to the problem at hand. But before we go into that, let's solve this the old fashioned way.
Procedure
Let n = the number of balls in the lottery and therefore the highest possible number that you can choose.
Let x = the number of picks that must be made correctly to win.
Since you have chosen 6 numbers, the chances of getting one of your 6 numbers number correct out of 50 is:
Now let's take a step up to see the chances of getting two of the six picks correct. The odds of getting the first pick do not change. You still have that same chance, but the odds of getting two numbers right increases quite a bit. To figure out the chances of getting the second number, you have to consider that you now have one less ball, and one less pick left to match. You now have a 5 in 49 chance of getting that second pick alone (1 in 9.8). Unfortunately, that is very much related to your previous pick. It is not a simple matter of getting each pick independently of one another. Statistically, the chances are multiplied for each pick that must be made because you have to get both of the numbers:
Now those odds are a little bit tougher now, aren't they? Logically, you may see the progression as the odds for each pick becomes higher and higher individually. Your odds of picking the final ball are 1 in 45 (remember that you started at 1 in 8.333 for the first ball). Take each individual chance of a correct pick and multiply it by each one of the others. This combined with the odds of getting all of the picks correct generates the following calculation:
So if your state increases in population, and/or you have people winning too often, then you may notice that they add an extra ball to the lottery. Redo the calculations above and notice the difference that adding one ball to the lottery can have on the overall odds of winning. Keep in mind that every entry is another dollar taken in by the state.
This is why some states also have a "Powerball lottery" that is shared with other states. Since the population is higher when combining the potential audience of multiple states, the powerball allow some control over the probability. The calculation is based on the same principle, but instead of your final pick being a 1 in 45 chance (still using the example earlier) it is now a 1 in 50 chance (assuming the powerball goes up to 50). Since you are only picking five balls from the original pool, you also only get a 5 in 50 probability to start with (which is 1 in 10 for your first pick compared to the 1 in 8.33 in the previous example). When you multiply that new equation out, you see the following:
By adjusting how high the powerball can be, the probability can be predicted much better. Recalculate the odds with a powerball of only 30 and notice the difference.
Application
Earlier I mentioned the term "factorial." I also mentioned that the order of the picks was unimportant. Because of this, there is a special rule that can be used to calculate the probability using factorials. This lets you use a calculator and save a lot of time. This is a special case called a binomial coefficient. A binomial coefficient has a special formula and notation that can be used to calculate the same probability. It is as follows:
n! nCx = ---------- (n-x)!x!Again, the same assumptions earlier are in force. "n" is still the number of balls and "x" is the number of picks. Our friend the factorial helps us out here. In our case:
50! 50! 50C6 = --------- Can be reduced to: ------- (50-6)!6! 44!6!Now, you may have to look at this closely, but remember the definition of a factorial and you can reduce this formula even further based on the logic and understanding of what a factorial is. 50! means 50 * 49 * 48 * ... and 44! means 44 * 43 * 42 * ... correct? Well, 50 is obviously larger than 44. Once you get to ... 44 * 43 * 42 ... you are going to be overlapping numbers in the denominator, or bottom of the equation! Since basic algebra tells you that a 44 in the numerator will cancel out a 44 in the denominator, the same holds true for factorials. In the following equation, the 44! in the numerator and the 44! in the denominator can be canceled out:
50 * 49 * 48 * 47 * 46 * 45 * 44! 50 * 49 * 48 * 47 * 46 * 45 --------------------------------- Leaving --------------------------- 44! * 6! 6 * 5 * 4 * 3 * 2 * 144! is the same as writing out all of the numbers on the bottom and crossing them out with all of the numbers on the top. We recognized ahead of time that this would happen and saved ourselves some time and space. You can write them out if you feel more comfortable visualizing the whole thing, but you will be using a lot of paper.
Now you find yourself looking at a simple multiplication and division problem. Calculate the equation the rest of the way out, and what number do you get? I'll bet that it is 15,890,700. And you can easily calculate the factorial portion of these equations on your trusty scientific calculator. The really good ones include the binomial coefficient formula built in and you simply enter the "n" followed by the key and then the "x" and magically your answer appears! It is not magic, it is mathematics.
Myths
O.K., so you want to try and "trick" the system and increase your odds. Unfortunately, you can't trick statistics and you can't trick mathematics. One of the more common tactics that I see people trying is to combine their money together as a group, usually at their job, to increase their chances of winning. On the surface it looks like you are increasing your odds of winning by having 20 chances to win instead of just one. Technically, it is a true statement. Unfortunately, it is a negligible amount of an increase compared to the loss you would get by splitting the money with your co-workers.
Method of number choice is another point of question. Does it help to pick your birthday and the birthdays of your family? What about auto-picks from the register. Are those more likely to win? Or less likely to win because the machine is "fixed?" Should you stay away from patterns like 1, 2, 3, 4, 5, 6 and scatter your numbers across the board? The answer is simple. Since history has no effect on picks, and since logistically the machines, balls, and people are verified by an independent accounting firm, the picks cannot be "rigged." All numbers have an equal chance of coming up at any given time.
Some people think that there are patterns that emerge in the lottery picks. They think that some balls simply have a tendency to occur more than others. This is simply not true. Individual numbers picked during the lottery change, but the chances of numbers over the career of the lottery will remain constant. Many lottery sites post historical picks for people to look for patterns or analyze the hell out of the numbers. This is all smoke and mirrors. They are perfectly happy to provide these numbers because they know that there is no pattern. If it convinces people to play more using their "pattern conspiracy theories," they will happily allow you to mislead yourself.
Did you really think you were the first to think of the old "play every combination" trick? Let me remind you that you would need almost 16 million dollars to play every combination! Even if you could somehow convince a bank or someone to back you on that bet, I pose two questions: Why would they need you when they could do it themselves? And what if someone else actually gets lucky and you have to split it with someone else? Oops! Don't forget about the government and the tax people!
Summary
The lottery, like most casino games, are fixed. I do not mean to say fixed as in, "they are cheating," but fixed statistically. Statistics are analyzed long before it is ever introduced. They know the odds, and they know how often they will win and how much they will make compared to how much they will have to pay out. The lottery will always, in the long run, benefit the states. They cannot lose. I know that is not what you expected to hear.
So how do you hack the lottery? I can sum up the answer to this question in two words. "Don't play." The only time the lottery was "hacked" was in 1980 in Pennsylvania and it involved tampering with the mechanics of the game, something that is now very controlled. If you are still interested in this story, you can look it up on the Internet quite easily. Keep your hard earned money in your pocket and don't let them take it from you under some false dreams of winning. If you play the lottery, they actually hacked you.
Shoutz: my statistics professors, all DDP members, everyone who has any part in the Binary Revolution at binrev.com.