At the beginning of the summer, I went to a debate camp where we talked a lot about some of the most common logical inconsistencies in debate. The one that stuck with me the most was this one called the “Gambler’s Fallacy“. This logical error is based on the idea that as long as you do something enough times, the desired result will occur. If you’ve noticed from its name, this also happens to be the main reason why people get addicted to gambling.
This idea that if I roll the dice one more time, the number I want has to show, simply because it hasn’t happened yet. But, the lottery system that we’ll be taking a look at today is a little bit different and I’ll go over why later in the article. Before I go any further however, I do need to warn you that there will be some math involved in the explanations. So, if you want your summer break to remain pure and untainted by math, this isn’t the place to be.
You’ve probably all learned or will learn about probability in your math class. The most important aspect of probability to us is the idea of dependent and independent probability. Dependent probability is when events are reliant on the outcomes of other events for their probability. For example, if everyone took a fruit out of a basket and there was only one apple among nine other fruits. The first person has a 1/10 chance of getting the apple, if the first person doesn’t get the apple, the second person will now have a 1/9 chance and so on. However, the most successful games of chance aren’t based on this principle simply due to the fact that this means there will always be a winner, which isn’t really the point of a game of chance. The type of probability that we want to focus on is independent probability where previous results don’t affect the current probability rates. This is most commonly modeled in the form of dice. Let’s say you’re trying for a four, on your first roll, the chance is 1/6. You don’t get the four, so you roll again. But, just because you didn’t get a five the first time, the chance for you to get a five the second time doesn’t increase (ie. it becomes 1/5), instead it remains 1/6. Now, when I first learned of this, the biggest question I had was why people kept coming back, it was starting from scratch every single time. However, it isn’t completely true that repeated attempts at something won’t increase your chances of getting a desired result, in a sort of roundabout way it does. While your chance of getting a four on any given roll is only 1/6, the chance of you getting any of the other five numbers for a huge number of rolls is also decreasing. We know that statistically speaking, the chance of getting any one result multiple times in a row progressively gets smaller and smaller. This applies to undesired results as well, with our scenario your chance of getting a negative result on the first try will be 5/6, or around 83%
. That’s rather high, yet on your second attempt the chances of you getting a negative result again becomes approximately 69%. this will continue to decrease every time you redo the event. Once you reach around 30 tries, your chance of not getting the one desired result in any of your thirty tries drops below the 1% threshold. This principle is what keeps people going back to these games, it’s this natural probability that ensures there are winners in these ridiculously unfair games of chance.
Following this idea, the lottery all of a sudden doesn’t look quite as good. Sure, you often hear that you’re more likely to have something else happen to you than win the lottery. But let’s take a look at what exactly the chances one set of seven numbers give you to win. Each of the seven numbers can range from 1-49, meaning your chance to guess all seven correctly hovers around 0.000000000001%, or approximately 1 in 700 billion. Unfortunate, I know. Even worse, the idea mentioned above about the likelihood of negative results happening really doesn’t apply either. So, the lottery is really all luck then. That’s what I thought too for a long time, the chances are so steep that it has to be complete luck. But during this camp, when we talked about the Gambler’s Fallacy, our coach reminded us not to mix up events in which this principle didn’t apply. Games, such as the lottery, in which it’s possible to cover all of your bases. Maybe not feasible, but absolutely possible. The lottery is a system with a finite number of possibilities, there are only so many possible combinations for seven numbers chosen from 1-49; especially when the order doesn’t actually matter. Essentially, if you went out and bought every single possible combination, you are guaranteed to win. This got me thinking, with the pot in most major lotteries now well above the 40-50 million dollar range, would buying every single possible combination of numbers net you a net profit? This is where it gets a little tricky, having to calculate just how many unique combinations regardless of order can be formed in the lottery. if you’ve done some higher level math, you’ll instantly recognize this as a combination problem. I’m not going to g
o into detail about the equation itself, but if you’re interested, go Google the combination formula and there’s plenty of great sites to learn from.
For the purpose of this article, just know that the equation is: n! / r!(n-r)!, where n is the total choices to choose from and r is the number of choices we get. That’ll look something like this when we plug our lottery values in: 49! / [7! * (49-7)!]; through this we can determine the total number of possible combinations to be around 86 million. Seeing as lottery sets sell for 3 sets for $5, we can tell around how much money you’ll need to actually buy every possible combination. The amount totals to a whopping $143 million, much more than the jackpot of $50 million. There’s a reason the lottery is so profitable and this is it. I called the system perfect because it is. The lottery is quite literally able to have someone buy every possible combination, and still come away with nearly 3 times the profit. There is really no feasible way for the lottery system to actually lose any money. There’s no way around it, the only way for you to win money off the lottery is by sheer luck.
So if you’re ever thinking of trying your luck at the lottery, think of it like this. Do you feel you’re one in seven hundred billion?
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