It is no secret to today’s educated person that casinos and various gambling establishments calculate all their games so as to always be in a winning position and to make a profit. This is done very simply: a person needs to return a winnings that correlate with their bet less than their odds of winning.

Yes, anyway, even the most sophisticated mathematical models come down to one thing on average: if you bet $1 and are offered $1000, then your chance of winning is less than 1/1000.

There are no exceptions, unless someone specifically wants to give you money. Keep this simple rule in mind so you can always look at the situation rationally. Game theory evaluates any strategy in the same way: the probability of winning is multiplied by its size. Roughly speaking, the maths says that a guaranteed $1000 is like getting $2000 with a 50 percent chance. This principle gives you a rough comparison of different games against each other. Which is better: a million dollars with a 1/100 000 chance or $50 with a 1/4 chance? Intuitively it seems that the first proposition is more interesting, but mathematically the second is more advantageous.

Mathematically speaking, it is impossible to win at casinos, since the product of winning probability multiplied by payout for each strategy chosen is always lower than the bet already made.

But the reason people gamble is that winning is not just about the money; it is also about the emotion of the process – and even more so, winning. Unfortunately, we cannot penetrate into the inner workings of the lottery. But it is useful to understand at least the formal procedure of how the lottery is run.

The famous slot machines, for instance, are actually a bit of a cheat: The wheel the player sees has symbols of different value drawn on it, but everything is arranged so that the player thinks the chances of each symbol falling out are the same. In fact (mechanically in older machines, and with software in modern ones), behind every visible wheel is a real one, on which the valuable symbols are rare and the cheap ones are frequent.

“Open” lotteries are much fairer in this sense. In the US, a common format is when the ticket either contains a sequence of numbers or it is chosen by the purchaser themselves. 

The probability of a jackpot in any lottery is usually calculated using a single formula. But calculating the probability of, for example, closing a single line in the lotto is quite nontrivial and would take an entire article, and maybe more than one. So actually the chance of getting some money in the lottery is higher due to the fact that most lotteries have additional prizes in addition to the main prize. But we’ll focus specifically on the jackpot for simplicity of evaluation.

Suppose we buy a lottery ticket with a random set of numbers. During the draw, the same number of balls are pulled out, and if the numbers on them match the numbers on the ticket (in any order, that’s important!), we win. The probability of such a win is calculated as follows:

Probability of winning = 1 ÷ Number of combinations of balls.

The number of combinations without taking into consideration the order of the balls is called in mathematics the number of combinations and if you know and understand the formula for its calculation, you probably do not learn anything new from this article. If you are not a mathematician, it is easier to use an online service, such as this one. Such services (and the formula behind them) offer to set two numbers:

• n is the total number of possible choices for one item. In our case the item is a ball, and there are as many balls as there are numbers in the lottery, more about that below.
• k is the number of items in one sample. In our case it is how many balls the lottery draws and how many numbers are in the ticket (these values are assumed to be equal).

So, if we have a lottery with 5 balls drawn and a total of 50 balls with numbers from 1 to 50, then the probability of winning it will be equal to one to the number of combinations at k = 5 and n = 50, i.e:

1 ÷ 2 118 760 = 0,00005%.

Consider the more complex case of the popular American PowerBall lottery, in which the jackpot value exceeded a billion dollars. According to the rules, there is a basic sample of 5 numbers (1 to 69), and one additional number (1 to 26). You have to get a match for all 6 numbers to win.

It is easy to understand that the chance of getting the first set is one to the number of combinations at k = 5 and n = 69 (that is, 11 238 513), and the chance of “catching” the last ball is 1 to 26. To get everything at once, these odds must be multiplied, because the events must occur simultaneously:

(1 ÷ 11 238 513) × (1 ÷ 26) = 1 ÷ 292 201 338 = 0,0000003%.

In other words, if 300 million people buy tickets, one person will win. This shows why winning the jackpot often won’t happen at all: lottery organisers simply don’t print so many tickets that there will be a winning one among them.

A PowerBall lottery ticket, by the way, costs $2. To calculate the benefit that would pay for the purchase of the ticket, you have to multiply the ticket price by 292 201 338.

Taxes need to be taken into account (to find out what percentage of the declared sum the winner actually gets, usually about 70%). That means the jackpot must be a minimum of $850 million, and it happens in this lottery. How is it possible, since I said in the beginning that the winnings with such multiplication is always not in favor of the player?

The thing is, if the jackpot draw fails, it is passed on to the next draw, so money is accumulated for a while and ticket sales continue.

In an ideal situation, you should skip all games without buying a ticket and then buy for the exact game where the draw will actually take place. But there is no way of knowing in advance. You can, however, start buying tickets as soon as the jackpot amount becomes larger than the mentioned amount. In such a situation the game will be mathematically profitable.

You can also find out if it is more profitable to buy many tickets for a single game or to buy one ticket for many games? Let us think about it. In probability theory, there is the concept of unrelated events. This means that the outcome of one event has no effect on the outcome of another. For example, if you roll two dice, the numbers on them are unrelated: in terms of chance, one die has no effect on the behaviour of the other. But if you draw two cards from the deck, the events are related because the first card determines which cards remain in the deck.

A popular misconception about this is called player error. It arises from a person’s intuitive notion that unrelated events are related. For example, if a coin tosses heads many times in a row, we’re inclined to think that the odds of making tails will go up because of it, but in fact, they don’t; the odds are always the same.

Going back to the lotteries: the different games are unrelated events, because the sequence of balls is chosen anew. So the chances of winning any particular lottery have no bearing on how many times you’ve played it before. This is very difficult to accept intuitively, because every time a person buys a ticket they think, “I’m going to get lucky, I’ve been playing for ages!” But no, the theory of probability is a heartless thing.

Buying several tickets for one game increases your chances proportionally, because the tickets within the same game are related: if one ticket wins, it means that another ticket (with a different combination) definitely won’t win. Buying 10 tickets increases the odds by a factor of 10 if all combinations on the tickets are different (in fact, they almost always are). In other words, if you have money for 10 tickets, it’s better to buy them for one game than to buy one ticket for 10 games.

Learn more interesting facts about lotteries at https://yesplay.bet/betgames.