Lotteries and mathematics
Lotteries are games of chance that have been integral to societies for over 2000 years, often operated by governmental or sports entities, where participants purchase tickets for a chance to win various prizes, typically money. The mathematical principles of probability, particularly concepts like randomness and expected value, play a crucial role in understanding how lotteries function and how fair they appear to players. In most cases, the odds are stacked against players, with many tickets yielding minimal returns and high jackpots being exceedingly rare. For example, in games like Powerball, the odds of winning the jackpot can be as low as 1 in 195 million.
Despite their inherent disadvantages, lotteries attract large numbers of participants due to the allure of potentially life-changing winnings for a relatively small investment. The distribution of winnings in lotteries often benefits the organizing body, with a significant portion of ticket sales going toward operational costs and public funding, such as education or infrastructure. While strategies like buying more tickets can marginally increase winning chances, lotteries are designed to be random and do not favor any specific winning strategy. Understanding the mathematical foundations and probabilities involved can help demystify the lottery experience for players, shedding light on the often misleading notion of fairness in these games of chance.
Lotteries and mathematics
Summary: A successful lottery depends on assuring the randomness of its selections and maintaining the perception of fairness.
Lotteries, which can be thought of as games that involve a winner selected by chance, have played an important role in the development of societies for more than 2000 years. Lotteries can include those run by political bodies, like states, where the winnings are money, or those run by a sports entity, like the National Basketball Association (NBA) Draft Lottery, where teams get to select new members. The U.S. government runs a Green Card Lottery program and selects winners using a computer-generated drawing. In most lotteries, very few people win anything substantial, and the purchase of a lottery ticket usually amounts to an unfair bet, in that the price of a single ticket is less than the average payoff across all tickets.
Nevertheless, lotteries are quite popular and consequently can raise substantial funds or allocate a small number of goods, services, or sought-after players among a large number of people or teams. The mathematical concepts of “randomness” and “expected value” are fundamental to the operation of lotteries and perceptions of fairness. Probability methods, especially combinations and permutations, are used to compute the odds or chances of winning, given certain conditions.
Distribution of Winnings
If lottery commissions somehow redistributed all of the ticket sale money into winnings for each game, then, at least in a cumulative sense, the purchase of lottery tickets would constitute fair bets—the average payoff would equal the average ticket price. An example of this would be if each player paid a dollar for a ticket that went into a hat, and then a winning ticket was chosen from the hat, with the purchaser of that ticket winning all of the money that had been collected. The reality is usually more complicated. Typically, multiple players can purchase the same ticket (thus having to share the winnings if that ticket is drawn) or the winning ticket might not have been purchased by anyone. In the latter case, the money is rolled over to the next game, which might be better than fair for the players if the jackpot is larger than the total investments for that week. Usually, however, the game is worse than fair for the players, primarily because the state (or whatever organization is hosting the lottery) keeps a portion of the proceeds. The state of Wisconsin, for example, pays out slightly more than half of its lottery revenue as winnings; most of the remaining revenue is used for property tax relief. Other common uses for funds among state-run lotteries include education, transportation, construction, and, ironically, help for compulsive gamblers.
Calculating the Chances
Regardless of the question of fairness, a lottery is clearly disadvantageous to almost every player. Nevertheless, lotteries attract large numbers of players because people are willing to pay a small amount of money for the small chance of winning a fortune. Powerball, operated by the Multi-State Lottery Association, provides a good illustration. There are nine ways to win with a $1 Powerball ticket; in four of these ways, the winnings are less than $10. The probability of winning something is about 1:35, but the probability of winning anything more than $100 is less than 1:700,000. The probability of winning the big jackpot is 1:195,249,054, as can be verified with some basic rules of counting.
Each Powerball ticket consists of five distinct numbers, 1–59, together with a “Powerball” number, 1–39. To determine the winning ticket, five balls are randomly drawn from a drum containing white balls numbered 1–59, and then one ball (the Powerball) is drawn from a drum containing red balls numbered 1–39. The winning ticket must match all five white balls (irrespective of the order in which they are drawn) as well as the red ball. The probability of winning the jackpot is 1 divided by the number of distinct possible tickets (the number of possible outcomes of the drawing). There are 59 possibilities for the first white ball; for each of those there are 58 possibilities for the second white ball. Continuing, there are 57 possibilities for the third, 56 for the fourth, and 55 for the fifth. If the order of drawing these balls were relevant, a total of 59×58×57×56×55=600,766,320 ways of drawing the white balls would be counted. This number, however, is much larger than the true probability, since the order of the drawings is not relevant. For instance, the possible outcome 2, 4, 8, 16, 32 should be counted once; but among the aforementioned count of 600,766,320, this collection of balls appears 5×4×3×2×1=120 times (because ball 2 could be listed in any one of five positions, and then ball 4 could be listed in any of the remaining four positions, and so on). The earlier count should be divided by 120 in order to correct for this systematic overcounting. Finally, incorporating the possibilities for the red ball, the result should be multiplied by 39. This calculation yields the 195,249,054 possible jackpot tickets.
Winning Strategies?
One way to improve the chances of winning is to buy more tickets. A properly run lottery does not lend itself to winning strategies. For instance, the Powerball drawings are videotaped and audited, and the equipment is stored in a vault and meticulously tested for nonrandom behavior. So bribery would be difficult, and knowledge of historical winning numbers would most likely be pointless. One could ensure a win by purchasing all possible tickets (an attractive option if the jackpot has grown very large because of rollovers), but this would require a huge initial investment, and it would be quite difficult from a practical standpoint to orchestrate the purchase. Further, if multiple people purchased the winning ticket, then the jackpot would be divided among them. Commonly chosen tickets involve previous winning combinations, numbers below 32 (because they could represent birthdays or other significant dates), and simple combinations such as 1, 2, 3, 4, 5, 6. The one bit of control a lottery player does have is to avoid such combinations to reduce the likelihood of splitting the jackpot in the event of a win.
Bibliography
Bialik, Carl. “Odds Are, Stunning Coincidences Can Be Expected.” Wall Street Journal (September 24, 2009). http://online.wsj.com/article/SB125366023562432131.html.
Hicks, Gary. Fate’s Bookie: How the Lottery Shaped the World. Stroud, England: The History Press, 2009.
North American Association of State and Provincial Lotteries (NASPL). “Cumulative Lottery Contributions to Beneficiaries.” http://www.naspl.org/index.cfm?fuseaction=content&PageID=74&PageCategory=74.