How to win the lottery

The famous "method" get rich quick — instant win lottery. As a rule, the chance to buy the winning ticket are very slim. All the more incredible is the story of New York workers deli Valerie Wilson, which won the top prize twice.

In 2002, she won a million dollars in the lottery Cool Million, having a chance to 1:5200000. Four years later, having taken part in the lottery Jubilee, she won the second million. At this time, the odds of winning were estimated as 1:7 056 00. What was her chance to win the grand prize in the two lotteries? It is easy to calculate: 1: (5200 000×705 600) = 1:3 669 120 000 000, or about one in 3.7 trillion. Chance, in fact negligible, as can be seen if we imagine another lottery.

Let us assume that it involves all the inhabitants of the earth — 7 billion people. Issued for the occasion 3.7 trillion tickets erasable protective layer, with only one winning, distributed equally among them. Under the rules of the lottery in the regular weekly draw each participant uses a single ticket. Of course, the first open ticket can be won. However, in a worst-case loss expectation of winning ticket will last for 10 years!

I must say, in the history of U.S. lotteries (who were many, and they are held constant), it was only a few cases with recurrent win prizes exceeding one million dollars. True, they are all related to conventional lottery jackpot which is usually much more of the main prize in the instant lottery. It is also noteworthy that one of the big wins — 254 million — went to in 2007, 84-year-old pensioner, whose name was … Wilson. So I do not believe after that match.

A record holder for the number of big wins was Joan Ginther
from Las Vegas. For 17 years, she has won in various lotteries four times — the last in 2010 — and the "earned" so more than 20 million dollars. According to the calculations, taking into account the available chances to win every time, such achievement is possible only in one case out of 36 x 1024, or 36 septillionov, against which even eclipsed 3.7 trillion, or 3.7 x 10??

How cleverly and correctly said about this one an American mathematics professor, an expert in the field of probability theory, this case is striking, but the probability of no memory! And characteristically, Joan chances to win in the next lottery are the same as any other player. In other words, there always are, even tiny. Must have before buying another ticket similar thoughts go (in fact — is encouraging), many fans of lotteries.

Six cherished

"Incredible happened" — commented the director of Bulgarian lottery "6 out of 42" falling in two consecutive identical circulation winning numbers: 4, 15, 23, 24, 35 and 42. These events occurred in September 2009, a difference of only four days. In a mere coincidence no one, including witnesses, did not believe him. However, careful inspection carried out under the personal supervision of the Minister of Sports, empowered to manage the lottery, the facts did not reveal fraud. Why all doubted that the same winning combination of numbers dropped twice by accident?

Let us make a simple calculation. All ways to choose six of the forty-two different numbers there Then the probability of winning the only possible set of numbers in every draw is

A serious doubt inadvertent discharge of the same numeric combination in two consecutive circulation caused by the fact that the probability of this event is very small, only

Note that in such a winning lottery repeated set of numbers sometimes still happens (in Bulgaria is the same, there were two or three times), though not in the two print runs in a row. But it is interesting that the organizers would take a lottery if the number had fallen twice, and even in the same manner?

Meanwhile, if the first time six numbers guessed wrong one, in the second they put 18 people at once! Hopes of those who hoped for a big win, not justified: had to share the prize money at all. As if to disprove the laws of probability theory, the day before the next draw many seriously discussed, do not put it on the same number again, because it is possible that they will fall for the third time!

While some players expect to win the lottery because supposedly they invented a system of guessing numbers (sooner or later it will, perhaps, give the result — by the same law of probability theory), the others are of the primitive strategies. Posed, for example, always on the same set of numbers and waiting when it falls (if they fall). Do that American Chris Hoffman, amateur lottery "5 out of 39", we can say lucky: I had to wait only 15 years old! About how much money he has spent in that time to buy tickets, and most importantly, it has covered all expenses winnings of \$ 150 thousand dollars, history is silent.

"Free" raffle

By the way, the money spent. In the mid-1990s in the U.S. was done "Postcode Lottery". It played out a cash prize of 5 million dollars, and it was a chance to get one of the 200 million from all participants were required to do: make a bid and send it by mail. Allowed to make as many bets, but with a condition — send them one by one. Whether to participate in such a "free" lottery?

Under such conditions, of course, is not necessary. Here is why. Knowing the probability of winning and the amount is easy to calculate the expected payoff of Player *:

However, the cost of sending letters to more! In other words, the expected return is much less than the amount invested player, and the difference is expressed by the number of negative. So if you spend money on sending multiple times, very soon have to count the losses.

This brings to mind the story of Anton Chekhov's "Life and adversity?" If his character immediately figured how much it will cost the winning ticket purchase home loan, who participated in the lottery money, then seriously think about, if he needed it at all.

Who will benefit from "Postcode Lottery"? The answer is obvious: the U.S. Postal Service. According to the most conservative estimates, the favorable developments revenue had sent letters to surpass the stated prize is at least ten times.

The People's Lottery

There are lotteries with material prizes. No wonder they are often called "people." Here is just one example. What did not come up with transporters to get passengers to pay for travel! In the spring of 2010 in Barnaul (Russia) conducted stimulating lottery "lucky ticket." It played out room and a series of single tickets for the tram and trolleybus. Prizes were given "in kind": gift cards electronics hypermarket, monthly ticket for public transport and the like. Needless to say, that in the three weeks of the lottery number of "rabbits" has declined sharply, and the income of Transport (if modest in general expenses) increased significantly!

Sometimes, the game itself captures nothing less than the desire to win a prize, especially when the possibility of winning built into each ticket. Is claimed, for example, the organizers of the momentary "All-housing lottery." And then, as it should be in advertising, "Do not believe me? Do not want to take the risk? Still not convinced? Then we go to you! "And not only so, but with a reasonable proposal to start practice on virtual lottery tickets.

Rules of the game are simple. The ticket nine lines (floors), two windows in each game, and in one of them hidden "lucky number", which is the sum of winnings (it is listed on the protective layer). The objective is simple — to guess the number, open only one window in the line. Thus, moving up from the bottom line, you can optionally open from one to nine windows. At first failure game stops, the ticket is non-winning. So how fair to the organizers? Clearly, the probability of choosing a "lucky number" for each step is the same — it is equal to one second — and as the game decreases exponentially:

And what is the average amount won per one ticket? If you make just one step out of the nine, it will be

and then will only decrease (due to the fact that the probability of winning is decreasing more rapidly than the increase prize money), but a loss, however, will increase.

No amount of train, and predict the outcome of the game in a particular installation. But to sit at the computer all the stands. If long enough to experiment with virtual tickets and record the results, we can see how to show himself a "law of large numbers" — as the number of trials the frequency of occurrence of the event (loss of large cash prize) approaches its probability, and the latter, as we have seen, is small.

With ticket in hand, wisely limited guessing numbers in the second and third lines and modest gains. Obviously, also judged many lottery participants. And what always count its organizers? As in any game of chance — on human weakness and inability to stop in time. Therefore, even on the tickets the next game "by popular demand players" was highlighted with a fire line amount (on the seventh floor), and add a new rule: if the fifth or sixth line will open a window with the symbol "+", the game is over, and you get a consolation prize — 50 rubles. Lottery organizers of ingenuity can not refuse, right? Do you think the rules are more honest and it increased after these innovations ticket sales?

Unforgivable blunder

Does the lottery organizers miscalculations? Do not believe it, but it happens. The following story took place in one U.S. state 20 years ago. The next card of the draw "6 out of 44" on sale for \$ 1, while the prize money was at that time of 27.9 million dollars, and almost all of them were in the jackpot.

Several investors realized that they can earn good money by buying cards and filling in all possible combinations of six out of forty-four numbers. They took into account the additional costs and risks (in particular by examining the statistics from previous draws, have found out that in 120 cases the winners were not available, 40 had one winner in 10 — two) and evaluated at the same time profit. In order to implement the planned 2.5 attracted thousands of small investors from different countries, as well as a group of people to work with the cards. As a result, because of the time was able to use only 70% of the planned number of cards, but it was enough to win the grand prize. But lucky!

What was the financial miscalculation organizers of the lottery, which prompted investors to strike a deal? And what were, by their calculations, the total expected profit and profit per dollar invested? (Minor costs can be ignored.) Miscalculation was that the organizers of the lottery for selling lottery cards disadvantageous price. In fact, it should increase by four times. Indeed, for a guaranteed victory to fill

cards. Tickets sold — 27.9 million dollars. So, the card should be worth at least 27900000: 7059052 = \$ 3.95. ** Not surprisingly, the price of 1 dollar seemed very attractive to investors. Ideally, the revenue from it is \$ 2.95. Calculations show that the total expected payoff is

Even with all the costs of revenue would have been huge!

It is easy to guess what the end of this story. After learning that the top prize went to a group of investors, the organizers of the lottery were initially reluctant to pay out winnings. But after some legal wrangling, were forced to admit that there is no reason to refuse. In the end, they had to pay dearly for their own mistake. An exceptional case, and quite instructive.

Persuasion

So whether to participate in sweepstakes, hoping to get rich quick? Is it worth taking a chance in the pursuit of easy money? Lucky or not — the big question. In the lottery regularly played by millions of people around the world and win one. But of one thing we can be sure — in any lottery organizers always win.

No reliable strategies of guessing the lottery numbers or identifying winning ticket in the instant lottery does not exist, but to fill out a card or buy up all the tickets. But this is possible only under the condition that players will join forces and make sense only if the prize money is much higher than the cost of the tickets, what can not count.

In the lottery of the "post" the organizers and players solve two inverse problems: first pick a combination that no one guesses, and the second guessing it carefully — and watch this "contest" is still more interesting than the occasional loss of numbers in lottery. What are the chances for success of both in theory? But in practice? Ironically, different: it is no secret that in any way unrelated parties often put on the same combination, and any kind of overlap and repetition in the arm to the organizers: almost always possible to calculate the most likely "circle" of such combinations.

Lottery organizers are interested in that it involved as many people. For an appropriate expected return and a reasonable price for the tickets of any prize money will pay off quickly due to the number of tickets sold (about what they share is a particular advantage, the players, of course, no one knows.) For sale and size affect the prize, and the probability of winning, and it can be varied by changing the rules of the game. So, if this probability is quite high (and in this case, the gain is always too small) or, on the contrary, is too small, the sale of lottery tickets is falling. In short, the success of the lottery depends on the ability of the organizers to find that "middle ground" between the value of prizes and a chance to win, which would provide them a large income. And this art mastered by them, and being familiar with the psychology of gamblers, and the basics of probability theory.

And finally, the main argument of reason, not only confirmed the centuries-old experience, but scientifically sound: in every game of chance each time luck and decide the case, but the more you play, the more you lose.

Glossary to Article

Combination — Selection of the elements of the n available (to less than or equal to n), where the order is not important. The number of possible combinations of n elements in k is denoted by and is given by

A random event — An event (the result, the outcome of the test), which objectively may or may not occur in this study.

The probability of event A
— A number expressing the probability of occurrence of an objective measure of a random event A, is defined as the ratio of the number of outcomes m testing conducive to the event (which lead to its occurrence), to the number n of all equally likely outcomes, that is equal

Random quantity — Variable X, which can take a certain value, depending on the case.

Expected value
(Or mean) of a random variable X, taking a finite number of values of x1. x2 ….. xn — a number equal to the sum of the products of these values in their corresponding probabilities p1. P2 ….. p, that is