It turns out that the difference between the different filling methods of the lottery forms is not only procedural
Danny Hellman
The big lottery draw for this year will take place tonight, and many are asking how to prepare to fill in the numbers on the form. Various ways of filling are open to gamblers, and contrary to what one might think, it turns out that not all of them are equal in terms of the odds.
In the big lottery, you have to guess seven different numbers between 1 and 30, in order to hit the seven winning numbers that will be drawn. A successful guess of four or more correct numbers wins a cash prize, and a perfect match between the seven numbers in the bet and those drawn in the lottery will result in the guesser winning the big prize.
One of the offered options is called "systematic lotto", in which the player purchases a package of guesses by filling in a single table. The participant marks on the designated form a set of eight to 12 numbers, thereby betting on all the sevens that can be formed from the numbers in the set. Thus, for example, a player filling out a systematic form 9 chooses a set of nine numbers, and in effect buys a bet on each of the 36 combinations of seven numbers that can be built from the nine numbers he has chosen.
A systematic lottery is offered to the guessing crowd as a convenient way to bet one on a number of combinations in a short and simple procedure. Another economical alternative is to automatically fill out the form using the computerized number generator, which produces random combinations of seven numbers for the bettor. However, the systematic game leaves the choice of numbers up to the purchaser of the form, a consideration that many do not underestimate.
However, even though a systematic lottery may look like a different filling procedure in the same game of chance, in practice it is a bet that differs in the distribution of chances from the alternative of choosing a series of random numbers. In fact, the chances of winning a cash prize with a systematic form are lower than the chances of winning with a form of the same price, which is filled out randomly.
To demonstrate this we will compare a systematic form 9 versus the alternative of betting on 36 series produced by the random number generator. There are 2,035,800 possible combinations of seven different numbers between 1 and 30, and each of them is an equal-probability candidate to win as the correct guess. The chance of each combination is therefore 1 in 2,035,800, and for any array of 36 different combinations the probability of containing the winning combination is 36 in 2,035,800 or 1 in 56,550.
It should be noted here that if the random filling is indeed random and blind, and there is no dependence between one combination and another, then the systematic form has a slight advantage in the chances of hitting the correct guess, because the lottery form may, with a probability of less than one-third per thousand, contain the same combination more than once, and in this case It will have less than 36 different combinations. This can of course be avoided if the random fill is reviewed to omit and replace any repeat occurrences, should they occur. On the other hand, it might be worth mentioning that in the prize system used in the lottery, in which the prize amount is divided equally among all the winners, there may be a reason to bet twice on the same combination: the story of a young couple who regularly guessed a certain series of numbers has already been published in Israel. One morning, the couple discovered that in the lottery held last night, their lucky series had been won, and that each of them individually carries a participation card on which the correct guess is marked. Besides them, there were two other winners, and thanks to the double eligibility, they pocketed half of the grand prize instead of a third - a difference of several million.
The difference between the systematic form and the lottery begins to become clear when you consider the chance of hitting six correct numbers out of the seven. The probability that the systematic form will contain a combination with at least six correct numbers is 1 in 1,131, while the probability that the lottery form will contain such a combination is more than three times higher and is approximately 1 in 350. This trend is even more pronounced when it comes to fewer shots. The probability of at least four correct numbers in at least one of the 36 series, i.e. the probability of winning some monetary prize, is lower than ten percent in the systematic form, and higher than seventy percent in the lottery form.
To get an idea of the origin of the difference in probabilities, it is useful to think about the particular case of winning the big prize, when the correct guess appears among the series included in the form. Now what is the probability that in the same form there will also be a combination in which six numbers are exactly correct? While for the remaining 35 random series it is a tiny chance of less than a third of a percent, for the systematic form it is an absolute certainty: the set of seven winning numbers contains seven subgroups of six correct numbers (such subgroups can be created by eliminating each of the seven winning numbers ). Any such subset can join any of the two remaining numbers in the systematic set, the two that are not part of the winning seven, thus creating a seven-number combination, included in the systematic form, in which six numbers are exactly correct. Whoever correctly guessed the seven winning numbers using systematic form 9, incidentally also guessed 14 combinations containing six correct numbers and another 21 combinations containing five correct numbers.
This grouping of the potential winning events, which is built into the systematic version of the game, is at the basis of the difference in the set of probabilities between the systematic form and the lottery. In the systematic game, the "success" of each combination is related to the success of its partners in the form, while in the lottery game the number of shots in one combination is independent of the number of shots in the other. The positive correlation between the number of shots among the systematic combinations, a correlation dictated by the great similarity in their composition, leads to a game with a probabilistic structure different from that obtained with independent guesses chosen at random. The winning scenarios open to the systematic player are crowded together in a limited area of the space of possible scenarios in the lottery, which leads to a considerable reduction in the probability of the prize.
One could perhaps argue that there is compensation for the reduced probability of winning, when looking at the expected profit expectancy, thanks to the high monetary prize awarded for systematic success. Indeed, every instance of winning through a systematic form is actually a multiple win. So, for example, four correct numbers within the set of nine numbers in the systematic bet, means ten different combinations in the same form, in which four numbers are correct, and accordingly - ten times the prize given in this class.
Even so, those who recognize the difference in the probability arrays may not be indifferent in choosing between the systematic option and the lottery one. What is it similar to? Imagine that you are visiting a casino, and standing in front of the roulette wheel. You have two discs in your hands that you can place on the betting table for the next spin of the wheel. Would you prefer to place both discs on the same number or on two different numbers? Concentrating the bet on a single number means a twice lower chance of winning a prize of double value. Tendencies of chance dispersion, such as those that find expression in diverse investment portfolios, might have led to a preference for a split bet. However, identifying the problem as such is not trivial at all when the question is embodied in the choice between different participation forms placed on the lottery stand.
And yet, precisely in the case of the lottery bet, the consequences seem more significant than in the case of the roulette example: here not only are different cases of secondary prizes grouped together, but they are also grouped within scenarios of winning a higher prize. A prize of hundreds of units is surely negligible for someone who has just raked in millions, but it may well bring some satisfaction when it comes alone. Ticket buyers may prefer to leave these winning options separate.
The field of probability is considered to be quite stealthy anyway (even without the special factor of combinatorial dependence, which mixes here into a mass gambling stage), and what escapes the eye in the intricacies of probability theory can sometimes increase the chances of success.
The author is a doctoral student in the Department of Psychology at Tel Aviv University
