Mathematics will help you calculate the probability of winning and determine which is more profitable: buy 10 lottery tickets for one game or a ticket for 10 different ones.

In the American TV series "4isla" (Numb3rs), the main character is a mathematician who helps the FBI in solving crimes. In one of the episodes, he utters the phrase that the probability of being killed on the way for a lottery ticket is higher than the probability of winning the lottery. At the end of the article I will give a calculation related to this statement, but now I want to talk a little about the math behind massive gambling and how it can help slightly increase your chances.

## Rule 1. Assess the risks

It is no secret for a modern educated person that casinos and various gambling establishments calculate all their games in such a way as to always be a winner and have a profit. This is done very simply: a person needs to return the winnings, which are correlated with his bet downward in comparison with his chances of winning.

Yes, one way or another, even the most complex mathematical models on average boil down to one thing: if you bet 1 ruble, and you are offered to get 1,000 rubles, 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 to always take a sober view of the situation.

Game theory evaluates any strategy in the same way: the probability of winning is multiplied by its size. Roughly speaking, mathematics believes that getting 1,000 rubles guaranteed is like getting 2,000 rubles with a 50% chance. This principle gives you the ability to roughly compare different games with each other. Which is better: a million dollars with a 1 / 100,000 chance or 50 dollars with a 1/4 chance? Intuitively, it seems that the first sentence is more interesting, but mathematically, the second is more profitable.

If you stay within the framework of only mathematics, you can calculate: it is impossible to win at the casino, because any chosen strategy leads to the fact that the product of the probability of winning by the size of the payment for the player is always lower than the bet he has already made.

However, people play because the gain for them lies not only in money, but also in emotions from the process - and even more so from victory.

And also because money for us is nonlinear: formally getting 1 ruble right now is like getting a million rubles with a chance of 1 / 1,000,000, but in fact, the loss of the ruble will not affect our condition in any way, nothing will change in life, but getting a million is a very serious event.

## Rule 2. Play in the open

Unfortunately, we cannot penetrate the inner kitchen of the lottery. But it is useful to understand at least the formal procedure of exactly how the draw is going.

For example, the famous slot machines "One-armed Bandit" and other slot machines are actually a bit of a trick: symbols of different values are drawn on the wheel that the player sees, but at the same time everything is arranged so that the player thinks that the chances of each symbol falling out the same. In fact (in old machines - mechanically, and in modern ones - with the help of a program) behind each visible wheel is hidden the present, on which valuable symbols are rare, and cheap ones often.

The chances of getting 777 on a slot machine are lower than the probability of getting any three cherries, and the difference can be tenfold.

"Open" lotteries are much more honest in this sense. In the United States, the format is widespread when the ticket either contains a sequence of numbers, or it is chosen by the buyer himself. In Russia, for example, the lotto format is preferred: there are several lines of numbers on the ticket, and you need to close either one of them (regular victory), or all (jackpot). In theory, a lottery company can "specially" print and sell non-winning tickets, and then manipulate the order of the balls, but in practice large companies do not do this: the organizers of the lottery always win, and the scandal in the event of the opening of bad faith will be huge.

If you intend to gamble, it is helpful to understand its mechanics and make sure there is no stakeholder influence on the results.

## Rule 3. Know your chances

The probability of a jackpot in any lottery is considered, as a rule, one formula. But calculating the probability, for example, to close at least one line in the lotto is very nontrivial and would take an entire article, or maybe more than one. Therefore, in fact, the chance to get some money in the lottery is higher due to the fact that most lotteries have additional prizes in addition to the main one. But I will focus on the jackpot for ease of evaluation.

Let's say we bought a lottery ticket with a random set of numbers. During the drawing, the same number of balls are drawn, and if the numbers on them coincide with the numbers on the ticket (in any order, this is important!), Then we won. 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 account the order is called in mathematics the number of combinations, and if you know and understand the formula for calculating it, then you most likely will not learn anything new from this article. If you are not a mathematician, then it will be easier to use an online service like this one. Such services (and the formula underlying their operation) offer two numbers:

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

So, if we have a lottery with 5 balls drawn, and there are 50 balls in total in the lottery with numbers from 1 to 50, then the probability of winning in it will be equal to one to the number of combinations for k = 5 and n = 50, that is:

1 ÷ 2 118 760 = 0, 00005%.

Let's consider a more complicated case - the popular American PowerBall lottery, in which the jackpot value exceeded one billion dollars. According to the rules, there is a basic sample of 5 numbers (from 1 to 69), as well as one additional number (from 1 to 26). You need to match all 6 numbers to win.

It is easy to understand that the chance of getting the first set is equal to one to the number of combinations for k = 5 and n = 69 (that is, 11 238 513), and the chance of "catching" the last ball is 1 in 26. To get everything at once, these chances need to be multiplied because the events must happen at the same time:

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

In other words, if 300 million people buy tickets, then only one will win. This shows why the jackpot is often not won at all: lottery organizers simply do not print so many tickets for a winning one to be caught.

## Rule 4. Start on time

The PowerBall lottery ticket, by the way, costs \$ 2. To calculate the benefit that would pay off the purchase of a ticket, you need to multiply the ticket price by 292 201 338.

Learn more about calculations. This is a reference to the first point, which says that the benefit of a solution is equal to its value times the probability. If we have an event with a probability of 1 / X and a value of N, then the benefit will be N / X. We spend \$ 2 and can calculate how much the winnings would pay off the purchase of a ticket:

• 2 = N ÷ X.
• N = 2 × X, and X here is just equal to 292 201 338, as shown by calculations from the previous part

You also need to take into account taxes (find out what percentage of the declared amount will actually go to the winner, usually about 70%). That is, the jackpot must be at least \$ 850 million, and this happens in this lottery. How is it, I said at the beginning that the gain with such a multiplication is always not in favor of the player?

The fact is that if the drawing of the jackpot did not take place, then it goes on the next time, and therefore the money accumulates for some time, and ticket sales continue.

In an ideal situation, you should skip all the games without buying a ticket, and then buy exactly for the game in which the draw will actually take place.

But it is impossible to know this in advance. However, you can start buying tickets as soon as the jackpot is larger than the mentioned amount. In such a situation, mathematically, the game will be beneficial.

You can also understand what is more profitable: buy many tickets for one game or buy one ticket for many games? Let's think about it.

In probability theory, there is the concept of unrelated events. This means that the outcome of one event does not in any way affect the outcome of another. For example, if you roll two dice, then the falling numbers on them are not related to each other: from the point of view of randomness, one dice does not affect the behavior of the second. But if you draw two cards from the deck, then these events are connected, 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 idea of the connectedness of unrelated events.

For example, if a coin comes up heads many times in a row, then we tend to believe that the chances of getting heads because of this will increase, but in fact this is not the case, the chances are always the same.

Returning to lotteries: different games are unrelated events because the sequence of balls is re-selected. So the chances of winning any particular lottery do not depend on how many times you have played it before. It is very difficult to accept intuitively, because every time a person buys a ticket, he thinks: “Well, now, you’ll be as lucky as you can, I’ve been playing a lot of time!” But no, probability theory is a heartless thing.

But buying several tickets for one game increases your chances proportionally, because the tickets within one game are linked: if one wins, then the other (with a different combination) will definitely not win. Buying 10 tickets increases the chances 10 times if all the combinations on the tickets are different (in fact, it is almost always the case). In other words, if you have money for 10 tickets, it is better to buy it for one game than buy it with a ticket for 10 games.

After your clarifications in the comments, it is fair to say that the probability of winning at least one game in a series of N games is higher than the probability of winning in any one particular game. However, it is still slightly less than the chances of winning by buying N tickets for one game, but the gap is quite small.

If you just take a ticket from your salary once a month for the sake of gambling, then, most likely, the very process of the game matters to you. Mathematically, it is more profitable to save up this money and buy 12 tickets at once at the end of the year, although, of course, losing in such a situation will be perceived more crushingly.

## Rule 5. Stop on time

And finally, I want to say that even the probability of 1/100 from the point of view of an individual is very small. If you check this probability once a month, then you will make 100 such checks in 8 years. Imagine how many times the probability is 1 / 1,000,000 or 1 / 100,000,000? Therefore, always bet only the amount that you are not afraid to lose completely, and not a ruble more.

In conclusion, as I promised, I will give an assessment of the statement from the beginning of the article. These data are for the United States, because the statement was formulated specifically for this country, besides, we have already calculated the odds for the American lottery above.

According to statistics, in 2016 in the United States there were about 17,000 murders committed in the United States, we will consider this as an average figure. And also suppose that a person is a potential target for murder when he is already an adult, but not old - that is, about 50 years during his life. This means that in these 50 years about 850,000 murders will be committed. The population of the United States is United States Population 325.7 million, so the chances of being included in a random sample of 850,000 are:

850 000 ÷ 325 700 000 = 1 ÷ 383 = 0, 3%.

But wait, this is just a chance to get killed. Namely, on the way to get a lottery ticket? Suppose you leave home for work every weekday, go out on one weekend, and stay at home the next. The average is 6 days a week, or about 26 days a month. And once a month you buy a lottery ticket. Therefore, the numbers obtained must also be divided by 26:

(1 ÷ 383) ÷ 26 = 1 ÷ 9 958 = 0, 01%.

And even with such a rough estimate, this is significantly more likely than a win. More precisely, it is 30,000 times more likely. In fact, of course, the numbers will be different: a person is endangered not only on the street, some people risk more than others, women are killed almost four times less often than men. But the principle is as follows.

Although living without faith in good events and with the constant expectation of bad ones, even knowing mathematics, is not the best choice.