Free analysis of the numerical lottery (lotto). The paradox of lotteries, or programs for selecting numbers How to calculate statistics of numbers falling out

Many people use various techniques and programs in the hope of winning a large sum in the lottery. But almost every one of these methods is based on faulty logic. After all, if significant programs for selecting a winning combination were freely available, then the lottery would completely lose its concept: all numbers are equally probable.

What is the paradox of lotteries?

Developers of both Russian and foreign programs for selecting lottery combinations claim:
— programs are not a simple random number generator, but a powerful mathematical and analytical tool for those who play and want to win, based on statistical analysis;
— programs allow you to control the lottery game, and not guess, selecting the next combination;
— the software saves money by applying filters that eliminate unlikely combinations;
— programs analyze various types of probabilities based on previous draws.

Some of these programs are offered to lottery fans to purchase for a small amount. Paid systems have advanced functionality. For example, a customizable number generator, in which you can include a sum filter and “a mode for overlaying played combinations on top of each other to obtain alternative statistics.”

In addition, Gayle Howard’s book “Lottery Master Guide,” priced at $24.50, is very popular online. According to the author, this is the most complete and complete guide to lottery strategies and selection of number combinations. “You'll learn how to identify specific numbers for specific lotteries and won't waste any more money. After reading the guide, you will know the world's best methods to win lotteries. You will improve your luck with the help of knowledge and skills,” reads the summary of the book. In addition, it is claimed that 107 people have already become winners of various lotteries thanks to the management (the count of winnings has been kept since 1985).

Gayle is advised to choose even and odd numbers for her combinations. In addition, it is stated that if you play with six numbers, then their sum must be in the range from 106 to 170.

Unfortunately, no number matching program can guarantee an accurate hit. If developers claim otherwise and distribute software for a fee, then this is fraud. So far, not a single millionaire of the Russian state lottery has said that he used some kind of program to select numbers, especially one purchased on the Internet. You can increase your chances of winning, but in completely different ways. Statistics of Russian state lotteries, archives of draws with winning combinations - everything you need to win is provided for each participant on the Stoloto website absolutely free of charge.

Remember, the paradox of lotteries is that the probability of winning a particular ticket is small, but the probability of winning any ticket is one, that is, 100%. This means only one thing: combinations 1, 3, 6, 10, 12 and 15, 20, 22, 31, 36 are equally probable and can occur in any of the draws.

Statistics on the Stoloto website

Of course, you can use number matching programs for fun or as a new method of playing. But we still strongly discourage you from purchasing paid software. With this amount you can make, for example, several more bets, which will increase your chances in proportion to the number of tickets purchased. And you will find all statistical data on the website. To avoid becoming a victim of another scammer, read this.

In the “Draw Archive” for each Russian lottery there are statistics on numbers drawn both for the entire time and for the last 10 draws:

An example of statistical data for the Gosloto 5 out of 36 lottery

Russian Lotto lottery statistics

Also, after registering on the site, each participant has the opportunity to estimate the number of occurrences of each number (the picture shows a graph of the occurrence of all numbers in the Gosloto “6 out of 45” lottery).

Frequently dropped pairs of numbers in the Gosloto “5 out of 36” lottery. Any number can be added to your bet.

In lotteries using the bingo system (Russian Lotto and Housing Lottery), a participant can select tickets either manually or by selecting “All numbers” from 1 to 90. In addition, in all lotteries you can use the “Favorite numbers” option.

And here is the combination that brought Igor S. more than 47 million rubles in Gosloto “5 out of 36”. Who could predict the probability that 2 pairs of numbers will follow each other? The answer was given by Igor himself: “I have my own way, which I follow. But I won’t reveal its secret.. When I think about what numbers to mark, I follow it from time to time. I look at frequently dropped numbers, for example. Why do I never bet big? I don't see much point in this. I believe that you can win with a small bet. You will either be lucky or not.”

Even if you take the time to study our statistics inside and out, you still will not have an absolute guarantee of winning. Winning the lottery is always a matter of chance, and no one can know the winning combination in advance. This is confirmed by our millionaires. Peter T. won more than 8 million rubles in the 2512th draw of Gosloto “5 out of 36”. The combination of 19, 5, 9, 35, 23 brought him success: “Over the years of participating in lotteries, I have tried many different schemes and formulas. I followed signs, kept track of lucky days, tried to find my lucky numbers, but it’s impossible to outwit luck. In the end, I won with completely random numbers.”

Andrey P., who won more than 6 million rubles in Gosloto 5 out of 36, says: “I choose the numbers according to how my hand falls and where my eye looks. I’m a cheerful person, and I’m not interested in calculating anything, I’d rather talk to my friends at this time.”

Two sisters from Murmansk, Tatyana and Lyudmila T., won a huge amount in Gosloto “6 out of 45” - more than 100 million rubles. And the secret of their victory is simple: “We buy lottery tickets on the eve of the birthday of one of our relatives. It was grandpa's birthday."

Natalya Kireeva won a million rubles in the Russian Lotto and explained her luck this way: “Everything happened spontaneously. A long time ago I saw a program on TV about lottery winners. And for some reason I remembered her when I walked past the lottery kiosk. She came up to him, then left again, as if something was pulling her. I took this attraction as a sign and bought a ticket. Then on Sunday I woke up two minutes before the start of the Russian Lotto program. Also a sign! Right up until the drawing itself, I was sure that I would win, even if it was a small amount. But, of course, I didn’t expect a million rubles!”

These examples are proof that in lotteries everything is decided by chance. And each of you has a chance to hit the jackpot. Therefore, you should not waste your time searching for programs on the Internet that provide “magic guarantees” or “predict combinations.” Under no circumstances should you be deceived if you are offered to tell you what numbers will appear in tomorrow’s draws, even for a small sum. We tell you with a 100% guarantee that only scammers do this. To be fully armed, read ours, and be vigilant!

Mobile application "Stoloto"

Your whole life is on the run and you don’t have time to go to a lottery kiosk? With ours, all problems will disappear overnight. After downloading it, you can purchase a ticket at any time, find out the results of previous draws, top up your Stoloto wallet and read about the latest news in the world of lotteries. The Stoloto application is available in two versions: for Android and iOS. Choose the version that suits your smartphone and use the most convenient and fastest way to buy lottery tickets.

In connection with the entry into force yesterday, June 30, 2009, of paragraph 1 of article 17, paragraph 1 of article 18 and article 19
FEDERAL LAW of December 29, 2006 N 244-FZ “ON STATE REGULATION OF ACTIVITIES IN ORGANIZING AND CONDUCTING GAMBLING AND ON AMENDMENTS TO SOME LEGISLATIVE ACTS OF THE RUSSIAN FEDERATION” (adopted by the State Duma of the Federal Assembly of the Russian Federation 12/20/2006), http://nalog.consultant. ru/doc64924.html

THE PARADOX OF THE LOTTERY AND BERNOULLI'S LAW OF LARGE NUMBERS

Opportunity - an opportunity to be disappointed

(“Aphorisms, quotes, and catchwords”,
http://aphorism-list.com/t.php?page=vozmojnost)

Your chances of winning the lottery will increase
if you buy a ticket

Winston Groom (from Forrest Gump Rules)
(“Aphorisms about games”,
http://letter.com.ua/aphorism/game1.php)

"The Lottery Paradox"

It is quite expected (and philosophically verifiable [English]) that this particular ticket will not win, but one cannot expect that no ticket will win” (“Academics”, List of Paradoxes, http://dic.academic.ru/dic.nsf /ruwiki/165304).

“The paradox of the lottery (such as sports lotto)

Most lottery players (in which the winnings are distributed among all the winners, as in sports lotto) usually do not bet on “too symmetrical” combinations, although all combinations are equally possible. The reason is simple. Players know from experience that, as a rule, non-symmetrical combinations win. In fact, it is more profitable to bet on the most symmetrical combinations precisely because... Why?" (excerpts from the book: G. Szekely. Paradoxes in probability theory and mathematical statistics. M.: Mir. - 1990, http://arbuz.uz/t_paradox.html).

SOLUTION

Everyone has played some kind of game in their life, not necessarily gambling, which is, in one way or another, related to probability. And if someone didn’t play, they probably tossed a coin a couple of times in their life. Just like that, for fun or when solving some issue on which it turned out to be overwhelming or impossible to make a choice on one’s own. And I did the same thing as a child. But even then, some doubt crept into my head about the correctness of justifying my choice of solutions to even trivial issues by tossing a coin. Apparently, even then I didn’t want to entrust my own right of choice to blind chance. But not so much because I myself can choose the best option right now and for myself, but more because such a choice will not be fair. So fair that without any further thought or internal hesitation I could accept it and act in accordance with this choice. And then I completely stopped further attempts at making decisions in such a simple way when my fears were confirmed while watching one of the popular Indian films that took place here in the 80s. If I'm not mistaken, it was the film "Revenge and Law." In it, one of the main characters, making a choice of something, tossed a coin with a serious look. And everything would have been fine, but only when he was shot anyway, and he gave him his “lucky coin”, it turned out that it had two identical sides. Apparently, this hero has well learned the first rule of success: if you want to win at a casino, become its owner.

To the question of the problem given by Székely in his book about why it is MORE PROFITABLE to choose symmetrical options for the geometric arrangement of numbers on the card field, the answer is not so complicated. The conclusion follows based on three conditions:

1) all options: both symmetrical and asymmetrical are equally probable;

2) most players choose asymmetrical options;

3) the amount of winnings received depends on the number of: a) participants, b) winners (according to winning categories, of course);

Therefore, from the point of view of benefit, that is, an increase in the possible profit when guessing, symmetric options will be guessed by a much smaller number of players with the same number of participants in the lottery, and the winning amount will be divided among a much smaller number of winners.

But on the other hand, if everything were that simple, then there would be no difficulties in determining the probability of certain events. And there are no fewer paradoxes and various paradoxical problems in probability theory, or even much more, than in other branches of science (in the same mathematics, logic, physics). For example, this task.

"The Dice Paradox"

A fair die, when thrown, has an equal chance of landing on any of the sides 1,2,3,4,5 or 6. (The sum of the points on opposite sides is 7, i.e. falling on 1 means rolling a 6, etc.) .

In the case of throwing 2 dice, the sum of the numbers drawn is between 2 and 12. Both 9 and 10 can be obtained in two different ways: 9 = 3 + 6 = 4 + 5 and 10 = 4 + 6 = 5 + 5. In the problem with three dice, 9 and 10 are obtained in six ways. Why then does 9 appear more often when two dice are thrown, and 10 when three are thrown? (excerpts from the book: G. Szekely. Paradoxes in probability theory and mathematical statistics. M.: Mir. - 1990, http://arbuz.uz/t_paradox.html)."

There is no paradox in this problem. The paradox, or rather the trick, is hidden in incomplete information: the number of possible combinations is greater than indicated. Because only the types of options are indicated, the methods of composition that need to be distributed over the number of bones.

The answer is simple: 9 appears more often when two dice are rolled, and 10 when three dice are rolled, because the probability of rolling a total of 9 with two dice is greater than the probability of rolling a total of 10 with three dice, which reflects the ratio of the number of options compilation of these amounts.

Number of options for summing up:

A. 9 on two dice: 3+6 (2 possible options, that is, on the first 3 on the second 6 and vice versa) and 4+5 (2 options). Total: 4 options

10 on two dice: 4+6 (2 var.) and 5+5 (1 var.). Total: 3 options

The odds ratio is in favor of the sum 9.

B. 9 on three dice: 1+2+6 (6 varieties), 1+3+5 (6 varieties), 1+4+4 (3 varieties), 2+2+5 (3 varieties) , 2+3+4 (6 var.), 3+3+3 (1 var.). Total: 25 options

10 on three dice: 1+3+6 (6 options), 1+4+5 (6 options), 2+2+6 (3 options), 2+3+5 (6 options), 2 +4+4 (3 options), 3+3+4 (3 options), 4+4+2 (3 options) Total: 30 options

The odds ratio is in favor of the sum 10.

Why does the probability of events give rise to so many contradictions?

I may be wrong, but in my opinion, even mathematicians, not to mention those who are not at all familiar with the theory of probability, are captive of one false initial premise about the probability distribution. This is the idea that events occur only according to their probability, without taking into account the distribution of probability over time. Life does not always go according to calculated patterns and exactly as it is described mathematically. A reflection of this two-facedness: mathematical calculation and at the same time not a coincidence with it, is given in the following paradox.

THE PARADOX OF BERNOULLI'S LAW OF LARGE NUMBERS

“The ratio of heads or tails to the total number of attempts with a large number of throws tends to 1/2. Some players believe that with a series of heads, the probability of landing tails increases. And at the same time, the coins have no memory, they do not know the previous throws, and each time the probability of heads or tails falling out is 1/2. Even if before that 1000 coats of arms fell in a row. Doesn’t this contradict Bernoulli’s law?” (excerpts from the book: G. Szekely. Paradoxes in probability theory and mathematical statistics. M.: Mir. - 1990, http://arbuz.uz/t_paradox.html).

Bernoulli's law of large numbers

“Let a sequence of independent trials be carried out, as a result of each of which event A may or may not occur, and the probability of the occurrence of this event is the same for each trial and is equal to p. If event A actually occurred m times in n trials, then the ratio m/n is called, as we know, the frequency of occurrence of event A. Frequency is a random variable, and the probability that the frequency takes the value m/n is expressed by Bernoulli’s formula ...

The law of large numbers in Bernoulli's form is as follows: with a probability arbitrarily close to unity, it can be argued that with a sufficiently large number of experiments, the frequency of occurrence of event A differs as little as desired from its probability, i.e....

...in other words, with an unlimited increase in the number n of experiments, the frequency m/n of event A converges in probability to P(A)" (Theory of Probability, §5. 3. Bernoulli's Law of Large Numbers. , http://www.toehelp.ru/ theory/ter_ver/5_3)

Thus, from the contradictions contained in these paradoxes, a general problem can be formulated.

Controversies:

1. The paradox of the lottery - the probability of winning a specific ticket is negligible, but the probability of winning any ticket is 1, that is, 100 percent;

2. The paradox of Bernoulli's law of large numbers - the probability of getting any option is equivalent, but in reality it should change as some options get out more to bring the probability to balance.

The problem, in my opinion, lies in the misunderstanding of the uneven distribution of probability over the number of options or, in other words, the dependence of the probability of one option of an event on another in a time context.

No one will argue that the sum of the probabilities of the event options is equal to one. But why does everyone think that the distribution among options is even? This approach completely ignores the variability of the world over time. And the same coin sides should then strictly alternate in turn: heads, tails, heads, tails. Then the probability distribution calculated by the formula will completely coincide with the actual one FOR ANY SPECIFIC TIME PERIOD. Because within this time period, the number of different options dropped will be the same. But in reality this is not the case. Within individual periods, the probability of each event option varies from 0 to 1 (from zero to one hundred percent). For example, when out of ten times, heads come up all ten times (or red, if it’s roulette in a casino). I know of a case where the roulette wheel came up black 15 times in a row. From the point of view of calculating probability, this is generally impossible if we take it as a unit, that is, the sum of all possible options, for example, 20 occurrences, which include these fifteen. And this, by the way, continuing the thought, for some reason did not lead to the next fifteen drops of red. Players call such hits in a row as streaks. Series are observed in sports, and everywhere in general.

Would you say that Bernoulli's law describes periods with large, "unlimited numbers of experiences" and within these limits it is true? Then why shouldn’t the same coin fall out first 1000 times on one side in a row, and then a thousand times on the other? After all, the law in this case is not violated one bit? In reality this does not happen. In fact, any long series of occurrences of two possible variants of events (A and B, which can be replaced, for example, by “heads” and “tails”) will closely correspond to the pattern of occurrences:

A, B, A, B, AAA, B, AA, BB, AA, BBBBBBB, AA, BBB, A, BBBBBBB, AAA, B, AA, BB, A, B, AAAA, B, AA, BBB, AAAA, B, A, B, A... (30 A and B each, 60 in total).

As you can see, within each specific segment (fallout periods or time periods) there are unevennesses. And the duration of the “series” of occurrences of one option a) in a row and b) within a period (for example, 10 occurrences) may fluctuate. Theoretically, the amplitude of such oscillations is not limited by anything, but there are no practically unlimited duration series. That is, there is a certain limit to which the duration of the “series,” its “length,” increases. These two restrictions regulate the balance of the probability of event options: firstly, the variability of options within an arbitrary period (time), in other words, the change in the “length” of series from 1 to several repetitions in a row, and secondly, the limitation of the length and frequency of series in within an arbitrary period (time). This achieves a variety of events, variability.

This probability distribution is noted by players who choose asymmetrical options for the arrangement of numbers on a lottery card. They do not proceed from an equal probability distribution for the number of numbers, that is, their equally possible occurrence, but precisely from an uneven probability distribution over the numbers. For some reason, the same numbers have not yet appeared, not only in two draws in a row, but in the mass of all draws. I can say this with confidence based on studying the “Sportloto 5 out of 36” lottery, which has been running for decades. In two draws in a row, a maximum of 1 number from the previous draw will appear (quite often - about a quarter of the draws), 2 (in isolated cases), 3 (in more rare cases). According to the theory of probability, someday all five numbers would come out the same for two draws in a row. But this would take thousands of years, even if the circulations were held every day instead of once a week. This follows if we assume that the total number of possible options in the “Sportloto 5 out of 36” lottery (36 * 35 * 34 * 33 * 32 / 1 * 2 * 3 * 4 * 5) = 376.992, and repeat five numbers of the previous draw will occur no earlier than all possible options have been drawn at least once, which will happen when conducting 1 draw per day, taking into account leap years for: 376.992 / (365 * 4 + 1) * 4 = 1032.1478 ~ 1032 of the year. But even after a complete search of all possible options in a row, two identical editions may not appear for several thousand years, and perhaps never.

Therefore, I absolutely agree with players choosing the most frequently dropped, asymmetrical options. Because waiting for the option to appear, for example, from the film “Sportloto - 82” with M. Pugovkin and M. Kokshenov - 1,2,3,4,5,6 is simply unrealistic. You might as well wait for rain on Mars.
I will add that, having fixed the probability distribution in a certain way, I saw that the types of options similar to those given from the film make up an insignificant fraction of a percent of all other types, classes of options that appear, and according to the theory of probability they are equally possible.

The paradox of the lottery arises due to the fact that the probability of winning each specific ticket separately, that is, any one, is negligible, tending to zero, but the probability of winning any one specific ticket is one hundred percent. Because the probability of specific numbers appearing in a specific draw is distributed unequally among all options. Roughly speaking, one hundred percent of the probability is divided not into the entire mass of tickets, but into two parts - all the winners (that is, one, for simplicity) and all the losers (all the rest). Thus, everyone and no one has a chance to win. Because it is impossible to know WHICH ticket will win, but we know in advance that SOME ONE ticket will win (without going into details of the number of winners and winning conditions).
At this point, no matter how funny it may seem, the correctness of “female logic” becomes obvious, which claims that the probability of a meteorite falling on Red Square is not one in several million, but fifty to fifty - either it will fall or not.
Apparently, such a famous mathematician as Poincare also held a similar opinion to mine. “Poincaré once remarked sarcastically that everyone believes in the universality of the normal distribution: physicists believe because they think that mathematicians have proven its logical necessity, and mathematicians believe because they believe that physicists have verified it with laboratory experiments” (De Moivre's Paradox , excerpts from the book: G. Székely, Paradoxes in probability theory and mathematical statistics (M.: Mir. - 1990, http://arbuz.uz/t_paradox.html).

That is, the lottery paradox arises due to an incorrect initial premise - the probability distribution is not uniform within a particular period, but variable. And if we take one circulation for a separate period, then ALL possible options CANNOT appear in it, but only ONE will appear. Therefore, the contradictory understanding of probability disappears: the probability of the absolute majority of options appearing will be equal to zero, and only the probability of one option will be equal to one.

There are no contradictory conditions in the lottery paradox:

1) only one option appears in a particular draw out of all possible ones (one ticket wins);

2) there are many more possible options.

Consequently, the probability of expecting to win only ONE of all possible options (tickets) tends to one, and the probability of expecting to win ALL REMAINING ONE options (tickets) tends to zero.

There is also no contradiction in Bernoulli's paradox of large numbers:

1) the probability of getting one of the possible options is half – 0.5;

2) the expectation of a change in the probability of the second of the possible options falling out after a series of falling out of the first one changes.

Consequently, the probability of the event as a whole does not change, that is, the sum of the probabilities of the options remains the same, but within a single period, especially if it is incomparably small in relation to the sum of all possible periods of occurrences, the probability changes, which is reflected in the expectations of the players.

Try to prove to the winner of a large sum that the probability of this was infinitesimal. Moreover, try to prove this to several or thousands of such people. The probability of even being born was absolutely negligible for some, but, nevertheless, it happened.
Many compare the impossibility of winning to the possibility of a meteorite falling on one's head or being struck by lightning. Try to prove that this is impossible, because the probability of this is infinitely small, to those affected by them. Like, for example, a woman who was healed from a lightning strike: “A unique case was recorded in the Serbian city of Slivovica, reports the DELFI portal. Lightning hit 51-year-old Nada Akimovich, who previously suffered from arrhythmia. However, as a result of exposure to a powerful discharge of electric current, the disease disappeared” (Lightning strike healed a woman/Dni.ru, 23:23 / 07/10/2009, http://www.dni.ru/incidents/2009/7/10/170321.html ) – or to a boy from Germany: “...The chance of being hit by a meteorite is 1 in a hundred million... “First I saw a large fireball, and then I suddenly felt pain in my hand.” (A German boy was hit by a meteorite / MIGnews.com, 06/14/2009, 02:42,

Thus, THERE IS NO CONTRADOX IN THE LOTTERY PARADOX, JUST IN THE PARADOX OF BERNOULLI'S LARGE NUMBERS.

01.07.2009 03:00 – 6.30

Photo - Gosloto, http://www.gosloto.ru/index.php?id=93

PS: the probability of another article appearing instead of this one was close to 100 percent, today or in the coming days. However, this did not happen. And the appearance of this article in the coming weeks was generally close to zero. However, it happened.

Reviews

"The chance of being hit by a meteorite is 1 in a hundred million... A German boy was hit by a meteorite." The example is not identical to winning the lottery, since it is not at all clear where the ratio “1 to one hundred million” comes from.

If we talk about the lottery, then, let’s say for Israel, winning the first prize is 1 in 18 million. The person who won knows that his chance was negligible, but he sees that people win at least once every month or two, and therefore, even “knowing”, he does not realize the “smallness” of his chance. The catch is that the chance is small only for a specific person, but for the country as a whole, with a population of 6 million, it is very logical to win one of 10-20 games (not everyone plays, but each player can fill out more than one form).
A classic scenario, as in the birthday paradox.

As for the numbers - not for me, I took the quote. And it’s not so important, in theory, that the numbers may not be entirely accurate, the main thing is that they illustrate the idea - even very rare events have happened, are happening and will always happen. Therefore, I think the example is still identical.

Yes, you yourself pleased with the numbers, Dmitry. Speaking about Israel, purely in Jewish terms, they reduced the country’s population a little, maybe by a couple of million :) And then why did you decide that the main prize is won “once or twice a month.” This is out of the blue, sorry. And don’t think that people are all stupid, that they don’t understand the insignificance of chance. They understand! But the costs compared to the profits are negligible, just as the chance of winning is negligible. So there is, one might say, a balance here. And some people actually win all their lives! I recently read about a woman who, after a health misfortune, began playing every quiz and lottery available. So her whole apartment is littered with various prizes. The guy often won the Russian Lotto with 1-2 tickets, when others received nothing even with a pack or two. I myself participated in the lottery at the presentation, where the 1st main prize - a computer - was won by a woman who bought a computer, that is, she had only 1 ticket-receipt. And the second prize - a monitor - was won by the guy who bought the monitor, also with the 1st ticket-check. There were a hundred or two people. However, fraud is also possible here, which is not uncommon in our country.

Well, there is no paradox. For one person, the probability of winning tends to zero, and for a country, it approaches one hundred percent. This is my conclusion. I talked about birthdays, but it is completely inadequate for this, as far as I remember. Suffice it to remember how they recruit for classrooms.

“they reduced the country’s population by a couple of million... why did you decide that the main prize is won “once or twice a month”. This is out of the blue, excuse me...” - about the number is true, due to my mistake I was using data for 2000, but about the “from the ceiling” - you’re wrong. It just so happened that for almost 5 years I worked as the head of the computer department of the Israeli lottery and all statistics went through the database I managed. The number of known users is updated every 10 years (so the data is from 2000), but the winnings and the number of winners with their amounts (even if it is only 10 shekels) are recorded twice a week. So this is not an assumption, but a statement.

“And don’t think that people are all stupid, that they don’t understand the insignificance of the chance” - I didn’t say that. My quote: “even though he “knows,” he does not realize the “smallness” of his chance.” A person is not able to comprehend very large or very small numbers, i.e. It is important for him to walk 10 km or 20 km, but the distance to the moon is 380 thousand or 400 thousand does not matter - he is simply not able to realize this, since he himself does not personally operate with such distances.
The odds can easily be reduced from 18 million to 1 to 9 million to 1 by just buying two tickets. A person imagines this as an incredible advancement. And it’s not about stupidity, but about awareness. In my memory, it’s rare... VERY RARELY that a person buys JUST ONE column in the lotto, precisely for this reason: to double, triple,...-10 times the chance. Although essentially it doesn't matter.

Ahh.. so it’s you Systematism and someone else there, then, sir? ok:) By the way, you didn’t respond to one of my old reviews, and God bless you. I forgot myself.

AS: having read to the words “for almost 5 years I worked as the head of the computer department of the Israeli...”, the reader automatically added “intelligence” and, either hiccupping or giggling, swallowed convulsively...#:-0))

As for increasing your chances: if you take 1-2 tickets, then count the increase as zero. If you start to really increase, the game will be at a loss, because there is no guarantee that everything will pay off in the end.

The daily audience of the Proza.ru portal is about 100 thousand visitors, who in total view more than half a million pages according to the traffic counter, which is located to the right of this text. Each column contains two numbers: the number of views and the number of visitors.

With very different rules, conditions of victory, prizes, however, there are general principles for calculating the probability of winning, which can be adapted to the conditions of a particular lottery. But first, it is advisable to define the terminology.

So, probability is a calculated estimate of the likelihood that a certain event will occur, most often expressed in the form of the ratio of the number of desired events to the total number of outcomes. For example, the probability of getting heads when tossing a coin is one in two.

Based on this, it is obvious that the probability of winning is the ratio of the number of winning combinations to the number of all possible ones. However, we must not forget that the criteria and definitions of the concept of “winning” can also be different. For example, most lotteries use the definition of “winning”. The requirements for winning the third class are lower than for winning the first, so the probability of winning the first class is the lowest. As a rule, this winning is a jackpot.

Another significant point in the calculations is that the probability of two related events is calculated by multiplying the probabilities of each of them. Simply put, if you flip a coin twice, the chance of getting heads each time is one in two, but the chance of getting heads both times is only one in four. In the case of three tosses, the chance will generally drop to one in eight.

Calculation of odds

Thus, to calculate the chance of winning a jackpot in an abstract lottery, where you need to correctly guess several dropped values ​​from a certain number of balls (for example, 6 out of 36), you need to calculate the probability of each of the six balls falling out and multiply them together. Please note that as the number of balls remaining in the drum decreases, the probability of getting the desired ball changes. If for the first ball the probability that the right one will come out is 6 in 36, that is, 1 in 6, then for the second the chance is 5 in 35 and so on. In this example, the probability that the ticket will be a winner is 6x5x4x3x2x1 to 36x35x34x33x32x31, that is, 720 to 1402410240, which is equal to 1 to 1947792.

Despite these scary numbers, people win regularly all over the world. Don't forget that even if you don't take the main prize, there are also second and third classes, which are much more likely to be received. In addition, it is obvious that the best strategy is to buy several tickets from the same draw, since each additional ticket increases your chances multiple times. For example, if you buy not one ticket, but two, then the probability of winning will be twice as high: two out of 1.95 million, that is, approximately 1 in 950 thousand.

The popular Megalot lottery requires the player to select and cross out 6 numbers out of 36. If the player matches several numbers, he is paid a win depending on the number of numbers guessed. It is extremely difficult to guess all the numbers, but systematically identifying 3-5 winning numbers is quite possible.

Instructions

Get ready for serious and systematic work. Determine in your family budget the amount that you can spend monthly on purchasing lottery tickets without harming yourself and your loved ones. Even if you don’t have the opportunity to regularly buy a ticket, you are obliged to watch all television draws and keep your statistics on them.

While watching TV shows with Megalot draws, collect statistical data on each of the numbers participating in the lottery. Consider how often each number is drawn and when it was last drawn. The more statistics you collect, the more accurate the information will be.

When choosing the numbers in which you intend to cross out, do so based on the statistical data you receive. Try to choose the numbers that appear most often and, preferably, those that have not appeared for a long time.

Do not trust statistical data obtained from the Internet or even from friends. In the first case, you will choose those numbers that are profitable

Hello!

My name is Ivan Melnikov! I am a graduate of the National Technical University “KhPI”, Faculty of Engineering and Physics, specialty “Applied Mathematics”, a happy family man and just a fan of games of chance. Since childhood, I have been interested in lotteries. I have always wondered by what laws certain balls fall out. Since I was 10 years old, I have been recording lottery results and then analyzing the data.

Sincerely,

Ivan Melnikov.

1. Mathematical odds of winning

   • Simple calculation with factorials

The most common lotteries in the world are games of luck such as “5 out of 36” and “6 out of 45”. Let's calculate the chance of winning the lottery using probability theory.

An example of calculating the possibility of receiving a jackpot in the “5 out of 36” lottery:

It is necessary to divide the number of free cells by the number of possible combinations. That is, the first digit can be selected from 36, the second from 35, the third from 34, and so on.

Therefore, here is the formula:

Number of possible combinations in a “5 out of 36” lottery = (36*35*34*33*32) / (1*2*3*4*5) = 376,992

The chance of winning is 1 in almost 400,000.

Let's do the same for a lottery like 6 in 45.

Number of possible combinations = “6 out of 45” = (45*44*43*42*41*40) / (1*2*3*4*5*6) = 9,774,072.

Accordingly, the chance of winning is almost 1 in 10 million.

• A little about probability theory

According to a long-known theory, each ball in each subsequent search has an absolutely equal chance of falling out compared to the others.

But not everything is so simple, even according to probability theory. Let's take a closer look at the example of tossing a coin. The first time we got heads, then the next time the probability of getting tails is much higher. If heads come up again, then next time we expect tails with even greater probability.

With the balls coming out of the lottery machines, it's about the same story, but a little more complicated and with a more significant number of variables. If one ball is drawn 3 times and the other is drawn 10 times, then the probability of the first ball being drawn will be higher than that of the second. It is worth noting that this law is diligently violated by the organizers of some lotteries, which change lottery machines from time to time. A new sequence appears in each new lottery machine.

Some organizers also use a separate lottery machine for each ball. Thus, it is necessary to calculate the probability of each ball falling out in each individual lottery machine. On the one hand, this makes the task a little easier, on the other hand, it complicates it.

But this is just a theory of probability, which, as it turns out, doesn’t really work. Let's see what secrets there are, based on dry science and statistical data accumulated over decades.

1. Why doesn't probability theory work?

   • Less than ideal conditions

The first thing worth talking about is the calibration of lottery machines. None of the lottery machines are perfectly calibrated.

The second caveat is that the diameters of lottery balls are also not the same. Even differences of the slightest fraction of millimeters play a role in the frequency of a particular ball falling out.

The third detail is the different weight of the balls. Again, the difference may not seem significant at all, but it also affects the statistics, and significantly.

• Sum of winning numbers

If we look at the statistics of winning numbers in a “6 out of 45” lottery, we will notice an interesting fact: the sum of the numbers that players bet on ranges between 126 and 167.

The sum of the winning lottery numbers for “5 out of 36” is a slightly different story. Here the winning numbers add up to 83-106.

• Even or odd?

What numbers do you think are most often found on winning tickets? Even? Odd? I can tell you with complete confidence that in the “6 out of 45” lotteries these numbers are equally divided.

But what about “5 out of 36”? After all, you need to choose only 5 balls; there cannot be an equal number of even and odd balls. So here it is. Having analyzed the results of lotteries of this type over the last four decades, I can say that slightly, but still more often, odd numbers appear in winning combinations. Especially those that contain the number 6 or 9. For example, 19, 29, 39, 69 and so on.

• Popular groups of numbers

For a lottery of the “6 to 45” type, we conditionally divide the numbers into 2 groups - from 1 to 22 and from 23 to 45. It should be noted that in winning tickets the ratio of numbers belonging to the group is 2 to 4. That is, either the ticket will contain 2 numbers from the group from 1 to 22 and 4 numbers from the group from 23 to 45 or vice versa (4 numbers from the first group and 2 from the second).

I came to a similar conclusion when analyzing the statistics of lotteries like “5 out of 36”. Only in this case the groups are split up a little differently. Let's designate the first group that includes the numbers from 1 to 17, and the second the one that contains the remaining numbers from 18 to 35. The ratio of numbers from the first group to the second in winning combinations in 48% of cases is 3 to 2, and in 52 % of cases – on the contrary, 2 to 3.

• Is it worth betting on numbers from past draws?

It has been proven that in 86% of cases, a new drawing repeats a number that has already appeared in previous drawings. Therefore, you just need to follow the draws of the lottery you are interested in.

• Consecutive numbers. To choose or not to choose?

The chance that 3 consecutive numbers will appear at once is very low, less than 0.09%. And if you want to bet on 5 or 6 consecutive numbers at once, there is practically no chance. Therefore, choose different numbers.

• Numbers with a single step: win or lose?

You should not bet on numbers that appear in the same sequence. For example, you definitely don’t need to choose step 2 and place a bet with this step. 10, 13, 16, 19, 22 are definitely a losing combination.

• More than one ticket: yes or no?

It’s better to play once every 10 weeks with 10 tickets than once a week with one. And also play in groups. You can win a big cash prize and split it between several people.

1. World lottery statistics

   • Mega millions

One of the most popular lotteries in the world was carried out according to the following principle: you need to choose 5 numbers out of 56, as well as 1 out of 46 for the so-called golden ball.

For 5 matched balls and 1 correctly named golden ball, the lucky winner receives the jackpot.

The remaining dependencies are shown in the table:

Statistics of dropped regular balls for the entire duration of the above lottery draws.

Statistics of golden balls drawn throughout the Mega Millions drawings.

The most frequently drawn combinations in the lottery are shown in the table below:

• Powerball lottery where more than a dozen lucky people have managed to hit the jackpot. You must select 7 main game numbers and two Powerballs.

1. Winners' stories

   • Lucky compatriots

Evgeny Sidorov from Moscow received 35 million in 2009, before that Nadezhda Mekhametzyanova from Ufa hit the jackpot of 30 million. “Russian Lotto” sent another 29.5 million to Omsk to the winner, who did not want to identify himself. In general, winning jackpots is a good habit of Russian people

• 390 million US dollars in one hand

In the lottery we already talked about, Mega Millions, a lucky winner who wished to remain anonymous won $390 million. And this is far from a rare case. In the same lottery in 2011, two people managed to hit the jackpot, which at that time consisted of an amount of 380 million. The cash prize was divided into two parts and awarded to people who guessed the winning numbers.

A pensioner from South Carolina decided to participate in the Powerball lottery and won 260 million, which he decided to spend on the education of his children, and also bought a house, several cars for the family, and then went traveling.

1. conclusions

So, here is a summary of the most effective rules, following which you are sure to win:

1. The sum of all the numbers you bet on on a lottery ticket must be calculated using the following formula:

Amount = ((1 + n)/2)*z + 2 +/- 12%

n – maximum bet number, for example, 36 in a “5 out of 36” lottery

z – the number of balls you bet on, for example 5 for the “5 out of 36” lottery

That is, for “5 out of 36” the amount will be like this:

((1+36)/2)*5 + 2 +/-12% = 18,5*5+2 +/-12% = 94,5 +/-12%

In this case, from 94.5 + 12% to 94.5 – 12%, that is, from 83 to 106.

1. Bet equally on even and odd numbers.
2. Divide all the numbers into two large groups in half. The ratio of the number of numbers on a winning ticket is 1 to 2 or 2 to 1.
3. Follow the statistics and bet on the numbers that came out in previous draws.
4. Don't bet on numbers with one step.
5. It’s better to play less often, but buy several tickets at once, and also get together with friends and relatives.

In general, be brave! Follow my rules, place bets, analyze statistics and win!

Is it possible to win the lottery? What are the chances of matching the required number of numbers and winning the jackpot or junior category prize? The probability of winning is easy to calculate; anyone can do it themselves.

How is the probability of winning the lottery generally calculated?

Numerical lotteries are conducted according to certain formulas and the chances of each event (winning a particular category) are calculated mathematically. Moreover, this probability is calculated for any desired value, be it “5 out of 36”, “6 out of 45”, or “7 out of 49” and it does not change, since it depends only on the total number of numbers (balls, numbers) and the fact how many of them need to be guessed.

For example, for the “5 out of 36” lottery the probabilities are always as follows

• guess two numbers - 1:8
• guess three numbers - 1:81
• guess four numbers - 1: 2,432
• guess five numbers - 1: 376,992

In other words, if you mark one combination (5 numbers) on a ticket, then the chance of guessing “two” is only 1 in 8. But catching “five” numbers is much more difficult, this is already 1 chance in 376,992. This is exactly the number (376 thousand) There are all kinds of combinations in the “5 out of 36” lottery and you are guaranteed to win it if you only fill them all. True, the amount of winnings in this case will not justify the investment: if a ticket costs 80 rubles, then marking all the combinations will cost 30,159,360 rubles. The jackpot is usually much smaller.

In general, all probabilities have long been known, all that remains is to find them or calculate them yourself, using the appropriate formulas.

For those who are too lazy to look, we present the winning probabilities for the main Stoloto numerical lotteries - they are presented in this table

Necessary clarifications

The lotto widget allows you to calculate the probabilities of winning for lotteries with one lottery machine (without bonus balls) or with two lottery machines. You can also calculate the probabilities of deployed bets

Probability calculation for lotteries with one lottery machine (without bonus balls)

Only the first two fields are used, in which the numerical formula of the lottery is used, for example: - “5 out of 36”, “6 out of 45”, “7 out of 49”. In principle, you can calculate almost any world lottery. There are only two restrictions: the first value should not exceed 30, and the second - 99.

If the lottery does not use additional numbers*, then after selecting a numerical formula, all you have to do is click the calculate button and the result is ready. It doesn’t matter what probability of an event you want to know - winning a jackpot, a second/third category prize, or just finding out whether it’s difficult to guess 2-3 numbers out of the required number - the result is calculated almost instantly!

Calculation example. The chance of guessing 5 out of 36 is 1 in 376,992

Examples. Probabilities of winning the main prize for lotteries:
“5 out of 36” (Gosloto, Russia) – 1:376 922
“6 out of 45” (Gosloto, Russia; Saturday Lotto, Australia; Lotto, Austria) - 1:8 145 060
“6 out of 49” (Sportloto, Russia; La Primitiva, Spain; Lotto 6/49, Canada) - 1:13 983 816
“6 out of 52” (Super Loto, Ukraine; Illinois Lotto, USA; Mega TOTO, Malaysia) - 1:20 358 520
“7 out of 49” (Gosloto, Russia; Lotto Max, Canada) - 1:85 900 584

Lotteries with two lottery machines (+ bonus ball)

If the lottery uses two lottery machines, then all 4 fields must be filled in for calculation. In the first two - the numerical formula of the lottery (5 out of 36, 6 out of 45, etc.), in the third and fourth fields the number of bonus balls is indicated (x out of n). Important: this calculation can only be used for lotteries with two lottery machines. If the bonus ball is taken from the main lottery machine, then the probability of winning in this particular category is calculated differently.

* Since when using two lottery machines, the chance of winning is calculated by multiplying the probabilities by each other, then for the correct calculation of lotteries with one lottery machine, the choice of an additional number by default is 1 out of 1, that is, it is not taken into account.

Examples. Probabilities of winning the main prize for lotteries:
“5 out of 36 + 1 out of 4” (Gosloto, Russia) – 1:1 507 978
“4 out of 20 + 4 out of 20” (Gosloto, Russia) – 1:23 474 025
“6 out of 42 + 1 out of 10” (Megalot, Ukraine) – 1:52 457 860
“5 out of 50 + 2 out of 10” (EuroJackpot) – 1:95 344 200
“5 out of 69 + 1 out of 26” (Powerball, USA) - 1: 292,201,338

Example calculation. The chance of guessing 4 out of 20 twice (in two fields) is 1 in 23,474,025

A good illustration of the complexity of playing with two lottery machines is the Gosloto 4 out of 20 lottery. The probability of guessing 4 numbers out of 20 in one field is quite fair, the chance of this is 1 in 4,845. But when you need to guess correctly and win both fields... then the probability is calculated by multiplying them. That is, in this case, we multiply 4,845 by 4,845, which gives 23,474,025. So, the simplicity of this lottery is deceptive; winning the main prize in it is more difficult than in “6 out of 45” or “6 out of 49”

Probability calculation (expanded bets)

In this case, the probability of winning when using expanded bets is calculated. For example, if there are 6 out of 45 in the lottery, mark 8 numbers, then the probability of winning the main prize (6 out of 45) will be 1 chance in 290,895. Whether to use expanded bets is up to you. Taking into account the fact that their cost is very high (in this case, 8 marked numbers are 28 options), it is worth knowing how this increases the chances of winning. Moreover, it is now very easy to do this!

Calculation of the probability of winning (6 out of 45) using the example of an expanded bet (8 numbers are marked)

And other possibilities

Using our widget, you can calculate the probability of winning in bingo lotteries, for example, Russian Lotto. The main thing that needs to be taken into account is the number of moves allocated for the onset of winning. To make it clearer: for a long time in the Russian Lotto lottery, the jackpot could be won if 15 numbers ( in one field) closed in 15 moves. The probability of such an event is absolutely fantastic, 1 chance in 45,795,673,964,460,800 (you can check and get this value yourself). This is why, by the way, for many years in the Russian Lotto lottery no one could hit the jackpot, and it was distributed forcibly.

On March 20, 2016, the rules of the Russian Lotto lottery were changed. The jackpot can now be won if 15 numbers (out of 30) were closed in 15 moves. It turns out to be an analogue of an expanded bet - after all, 15 numbers are guessed out of 30 available! And this is a completely different possibility:

Chance to win the jackpot (according to new rules) in the Russian Lotto lottery

And in conclusion, we present the probability of winning in lotteries using a bonus ball from the main lottery drum (our widget does not count such values). Of the most famous

Sportsloto “6 out of 49”(Gosloto, Russia), La Primitiva “6 out of 49” (Spain)
Category "5 + bonus ball": probability 1:2 330 636

SuperEnalotto "6 out of 90"(Italy)
Category "5 + bonus ball": probability 1:103,769,105

Oz Lotto "7 out of 45"(Australia)
Category "6 + bonus ball": probability 1:3 241 401
“5 + 1” – probability 1:29,602
“3 +1” – probability 1:87

Lotto "6 out of 59"(Great Britain)
Category "5 + 1 bonus ball": probability 1:7 509 579