Do you have any chance to win the lottery

Denis Peshekhonov

Of Education Master-techieIn life has been developing programs and sometimes builds mathematical models for games.

In the American TV series "4isla» (Numb3rs) the main character - a mathematician to help the FBI in solving crimes. In one episode he says the phrase that the probability of being killed on the way of a lottery ticket is higher than the probability of winning the lottery. At the end of this article I will provide the calculation associated with this statement, and now I want to talk a little bit about mathematics behind the massive gambling and how it can help a little to improve their chances.

Rule 1. assess a risk

For the modern enlightened person knows that the casino and various casinos expect all of their games so that always be a winner and to have profit. This is done very simply: man needs to return the prize, which is related to its stake in the smaller side compared to his chances to win.

Yes, one way or another, even the most complex mathematical models on average are reduced to one: if you bet 1 ruble, and you are offered to get 1000 rubles, then your chance to win - less than 1/1 000.

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There are no exceptions, unless someone specifically wants give you money. Keep in mind this simple rule is to always take a sober look at the situation.

Game theory evaluates any strategy is similar: Chance to get the win is multiplied by its size. Roughly speaking, the math says that is guaranteed to get 1000 rubles - this is how to get 2000 rubles with a 50 percent chance. This principle enables you to roughly compare the different games together. Which is better: a million dollars with a chance of 1/100 000, or $ 50 with chance 1/4? Intuitively, it seems that the first proposal interesting but mathematically profitable to the latter.

If you stay in one only mathematics can calculate: to win at the casino is impossible, since any selected strategy leads to the fact that the product of the probability of winning the payout for the player is always lower than the rate which he already done.

However, people play because a win for them is not just about money, but also in the emotions from the process - and even more so from the victory.

And yet, because money for us nonlinearity formally receive 1 ruble now - this is how to get a million rubles with a chance of 1/1 000 000, but in fact the loss of the ruble will not affect our state in life will not change absolutely nothing, but getting a million - a very serious event.

Rule 2. Play in the open

Unfortunately, to penetrate the inner workings of the lottery, we can not. But it is useful to understand at least the formal procedure of how it goes hoax.

For example, the famous slot machines "one-armed bandit" and other slot machines - it's actually a bit of trickery: on wheel, which sees the player, painted symbols of different value, but everything is arranged so that the player thought allegedly odds loss of each symbol are the same. In fact (in older machines - mechanically, but modern - with the program) for each of the visible wheel hiding now, where valuable symbols are rare, and cheap - often.

The chances of falling 777 on the slot machine is lower than the probability of getting any three cherries, with contrast can be dozens of times.

"Open" lottery in this sense, much more honest. In the US, widespread format, when the ticket is or contains a sequence of numbers or she is chosen by the buyer on their own. In Russia, for example, prefer bingo format on the ticket are several lines of numbers and needs to close, or one of them (a common victory), or all (the jackpot). In theory, conducting a lottery company can "specifically" to print and sell non-winning tickets, and then manipulate the order of balls, but practice, large companies do not: the organizers of the lottery and so always win, and the scandal in the case of bad faith will be opening huge.

If you intend to play in a game of chance, it will be useful to understand its mechanics and make sure there is no influence of stakeholders on the results.

Rule 3. Know your chances

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

Let's say we bought a lottery ticket with a random set of numbers. During the draw pull the same amount of balls, and if the number of them coincide with the numbers on the ticket (in any order, it is important!), Then we won. The probability of such a win is calculated as follows:

The probability of winning = 1 ÷ number of combinations of balls.

The number of combinations without regard to the order called in mathematics the number of combinations, and if the formula for its calculation you know and understand, that from this article, you most likely will not learn anything new. If you are not a mathematician, it will be easier to use an online service, for example Now this. These services (and the formula behind their work) offer to set two numbers:

• n - the total number of possible variants of the same subject. In this case, the subject - it's a ball and all the balls as much as the numbers in the lottery on that below.
• k - the number of items in one sample. In our case - how many balls lottery playing and how much at the same numbers on the ticket (assuming that these quantities are equal).

So, if we have a lottery with draw of 5 balls, and just 50 lottery balls with numbers from 1 to 50, the probability of winning in it is equal to one to the number of combinations for k = 5 and n = 50, ie:

1 ÷ 2 118 760 = 0,00005%.

Consider the more complicated case - American popular lottery PowerBall, wherein the jackpot value exceeded one billion dollars. According to the rules base is a sample of 5 numbers (1 to 69), and one additional number (1 to 26). We need to get match all 6 numbers to win.

It is easy to understand that the chance to obtain a first set equal to one to the number of combinations for k = 5 and n = 69 (ie, 11,238,513), and a chance to "catch" the last ball - 1 to 26. To get it all at once, the odds must be multiplied, because the events must take place simultaneously:

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

In other words, if 300 million people will buy tickets, will win some one. This shows why winning the jackpot often do not take place: the organizers of the lottery simply print as many tickets that among them were winning.

Rule 4. start time

The PowerBall lottery ticket, by the way, is $ 2. To calculate the benefit that would be paid for the ticket purchase, you need to multiply the price of the ticket on 292 201 338.

More should be taken into account taxes (Find out what percentage of the declared amount actually get to the winner, this is usually about 70%). That is the jackpot must be at least $ 850 million, and it happens in this lottery. How is it that I'm in the beginning said that winning in this multiplication is not always in favor of the player?

The fact is that if the jackpot draw did not take place, then it moves to the next time, and so for a while the money piling up, and ticket sales are continuing.

In an ideal situation, you need to pass all the game without buying a ticket, and then to buy it at that game, which really draw will be held.

But know this in advance is impossible. However, you can start buying tickets as soon as the size of the jackpot will be more of the sums. In such a situation mathematically game will be profitable.

More can be understood that more profitable to buy a lot of tickets for a single game, or buy one ticket for a lot of games? Let's think.

In probability theory is the concept of unrelated events. This means that the outcome of one event does not affect the outcome of the other. For example, if you throw two dice, the loss of numbers on them are not related to each other: in terms of accidents, one die does not affect the behavior of the second. But if you pull from the deck two cards, then these events are related, because the first card depends on what cards remain in the deck.

A popular misconception about this so-called - gambler's fallacy. It arises from the intuitive idea of ​​the human connectedness unrelated events.

For example, if the coin many times in a row drops eagle, we are inclined to believe that the chances of tails because of this increase, but actually it is not, the odds are always the same.

Returning to the lottery: different games - a unrelated events, because the sequence of balls is selected again. So the chances of winning do not depend on the number of times before you played it in any particular lottery. It is very difficult to accept intuitively because the people each time you buy a ticket, thinking, "Well, Now some are lucky, how can I have a lot of time playing "But no, probability theory - heartless thing.

But buying several tickets for one game increases your chances in proportion, because tickets in one game tied: if you win one, then the other (the other combination) is not exactly will win. Buying 10 tickets increases the chances 10 times, if all the combinations on different tickets (in fact almost always is). In other words, if you have the money for 10 tickets, it is better to buy them in one game, you buy a ticket for 10 games.

After your updates in the comments is fair to say that the probability of winning at least one game in the series of games N is higher than the probability of winning in any one particular game. However, it is still a little less than the odds of winning by buying N ticket for one game, but rather a small gap.

If you just payroll once a month ASSUME ticket for excitement, then, most likely, the value for you is the game itself. Mathematically profitable to save money and at the end of the year to buy 12 tickets at once, though, of course, the loss in this situation will be perceived more crushing.

Rule 5. time stop

And finally I want to say that even the probability of 1/100 from the point of view of an individual - it is very small. If you check this possibility once a month, 100 such checks do for 8 years. Imagine how many times lower than the probability of 1/1 or 1/100 000 000 000 000? Therefore always put only the amount that is not afraid of a total loss, and no longer ruble.

In conclusion, as promised, here is the statement of an opinion from the beginning of the article. These data for the US, because the statement was formulated specifically for that country, besides the above, we have considered the chances for the US lottery.

According to statistics, in 2016 the United States was committedCrime in the US - 2016 about 17,000 murders, we assume this average figure. And yet, suppose that a person is a potential target for murder, when he was an adult, but not old - that is about 50 years in the course of his life. So it is made about 850,000 murders during these 50 years. The US population isUnited States Population 325.7 million people, it has a chance to hit 850,000 the size of such a random sampling:

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

But hey, it's just a chance to be killed. Namely, the path for a lottery ticket? Suppose you leave the house to work every weekday, in one weekend somewhere to go out, while the other stays at home. On average, it turns out 6 days a week, or about 26 days a month. And once a month you buy a lottery ticket. So these numbers should be more and divide by 26:

(1 ÷ 383) ÷ 26 = 1 ÷ 9 958 = 0,01%.

And even with such a rough estimate is significantly more likely than winning. More specifically, 30 000 times more likely. In fact, of course, the numbers will be different: the person is in danger, not only on the street, some people are more at risk than others, women are killed almost four times less than that of men. But the principle is.

Although live without faith in the good things and the constant expectation of bad, even knowing the math - it's not the best choice.