Why the lottery defies the odds

The lottery remains the most popular game of chance in history. The reason: The magnitude of the potential prizes it offers.

The lottery is also very different from other regulated games of chance in terms of the operator's setup, the payout schedule, and the players' strategic approach. A striking characteristic of the lottery is that the chances of winning are the lowest among the classical games of chance, at least for the top prizes. They're in the order of one to millions or tens of millions. Yet the popularity of the lottery has been unaffected, and lottery enthusiasts persist despite astronomical odds.

In this article, we will explore and attempt to explain this behavior from a cognitive-psychological perspective in relation to the features of the lottery.

The lottery and casino games

A study carried out in 2023 found that the most common form of gambling in the United States was playing the lottery, with a prevalence of 37%, higher than the prevalence of casino gambling (Statista, 2024). Similar studies confirmed the prevalence of higher lottery gambling rates in other countries, such as the United Kingdom (Gambling Commission, 2024) and Australia (Yogonet Gaming News, 2024), compared to casino gambling.

The statistics are verifiable. Yet the question remains: Why?

The lottery is a very different game from classical casino games.

• In both private and state lotteries, game setup stretches over several days as players buy tickets.
• The payout schedule is not set up as pre-determined rates specific to each prize and applied to the wager (as in casino games). Usually, shares of the prize fund are divided among winners, to be determined after the ticket sales are closed.
• Players do not have strategies available for their play except predicting the eventual winning numbers and perhaps choosing their investment in tickets. There are casino games with even greater strategic limitations, such as slots, but the other two factors of distinction (setup and payout schedule) remain determinant overall.

Yet, such distinctions don't seem to argue for the lottery's popularity. On the contrary, in the lottery, there are no sparkling cases and piercing sounds like those found in casino games, nor social interaction and strategic involvement. By comparison, it would seem mundane. But the lottery offers big prizes – comparable to the jackpots of progressive slots – and as such, we have to give a special focus to this feature to answer our initial question.

Microscopic odds of winning

The big prizes offered in the lottery come at the cost of tiny odds of winning for most of the lottery matrices. When saying “tiny,” it's not like something measured in millimeters compared to something measured in meters or even miles but measured in tens of miles.

Mathematically speaking, the probability of winning a certain type of prize in a given lottery is a combinatorial probability; that is, it reverts to a combinatorial calculus. It is the ratio between the number of combinations of the size of the draw holding the number of winning numbers required for that type of prize and the total number of combinations of that size. However, the factors in this fraction are combinatorial and reflect a hypergeometric distribution.

This means that the denominator increases with the size of the draw and the total number of numbers in the urn at a higher rate than the numerator of that fraction (Barboianu, 2009). Besides, the denominator itself may be a huge number. For instance, in a “6/49” lottery, the sample space counts 13,983,816 combinations of numbers, which yields a probability of 1 to 13,983,816 for the top prize (6 numbers hit from 6 drawn). Other lotteries offer even lower odds for the jackpot, and second or third-tier prizes have very low odds, too. Hence, the astronomic number of combinations in the sample space turns into a microscopic probability of winning the top prizes.

Being aware of and interpreting the lottery odds

Most lottery players are aware of such figures, which are displayed on the sites of the majority of lotteries. If they are not, they are computable or retrievable from expert resources.

They say the odds of winning the lottery are lower than the odds of being struck by lightning or dying in a plane crash. I find such comparisons relevant. However, I do not fully endorse them. The reason is that the numbers provided as odds for such daily life events are actually relative frequencies provided by irrelevant empirical statistics and not pure computable probabilities like the lottery probabilities are.

I think the best way to perceive the odds of winning the lottery is by interpreting them in frequentist terms.

For instance, consider the odds of hitting five white numbers in Mega Millions, which are 1 in 12,607,306. Assume that you buy ten lines each time to increase your chances ten times, namely about 1 in 12,607, and play bi-weekly to catch every draw. Your odds to win in one week are now about 1 in 6,303, so you have to play more than 131 years constantly to meet the statistical average of one win over that cycle.

I won't do the math for the jackpot (which has a probability of 1 in 302,575,350 and raises the average period for winning proportionally), but I would ask how it is that lottery players are disposed to wait (more than) a lifetime for a big win? I am joking, of course, as they could win at the next draw, just as they could never win, but the answer is more complex than that.

Perception of the odds of winning the lottery

Before exploring the explanations for the fact that people persist in playing the lottery despite almost null probabilities of winning, we should talk a bit about how people perceive such probabilities.

Perception is a matter of each individual's cognition and neuro-psychological profile. However, there are things for which people share similar patterns of perception, and often such perception is not adequate. It is the case with probability concepts.

Probability can be very tricky for those unfamiliar with it. In some instances, it is tricky, even for the experts. Those of you who have learned about probability may have heard about the classical problems in recreational math, such as 'the birthday problem,' 'the boy-girl problem,' or the 'three-door problem.' They are just elementary examples of how mathematical probability can defy intuition when applied in the real world.

The probabilities associated with the lottery, although of a different nature, have their own heady flavor, which can easily lead to inadequate perception of their magnitude for both math and non-math people.

Indeed, even savvy players may not realize that a well-known game number matrix, like “6/49,” can generate an astronomical number of possible combinations and, hence, microscopic probabilities. For a math-inclined person, the mathematical explanation resides in the power of expansion of a combinatorial unfoldment, but even for them, the first impulse may be to believe that the probability may not be that low until a precise computation is executed.

Psychological factors explaining the persistent play against the odds

Since the odds of winning the lottery may be easily misperceived, misinterpreted, or overestimated, this factor -call it the mathematical factor - could itself stand as an explanation for the persistent play. However, there is evidence that gamblers can constantly play the games they enjoy, even without knowing the probabilities associated with those games. This is the case with slots, whose inner design is kept secret by their producers. Slots are at the top of the popularity of casino games, and when one plays a game with secret non-transparent characteristics, we may fairly assume that they take into account as an option that the winning probabilities in that game might be very low since they are kept secret. And they really are (Barboianu, 2014).

The slots case lessens the explanatory power of the mathematical factor. This is why this factor should be considered in conjunction with cognitive-psychological factors, some of them specific to the lottery, some submitting to problem gambling, and some more general.

The Gambler's Fallacy

The Gambler's Fallacy is a cognitive distortion occurring in every game of chance and is related to misjudging probabilities and other statistical concepts. In particular, some lottery players may think that if a number combination, set of numbers, or a number alone hasn't been drawn for a certain period, then it is “due” soon. They fallaciously believe that it is more likely to occur in the next drawings than its actual mathematical probability. The fallacy can also be generalized to include the general event of winning:

Since I've never won thus far, my chances to win in the future increase with every new losing draw.

In reality, each lottery draw is independent of another, and past outcomes do not affect future ones.

Availability heuristic and chasing the jackpot

People come to overestimate their chances of winning the lottery also because of what psychologists call the availability heuristic. This is a mental shortcut relying on immediate examples that come to a person's mind when judging a specific concept, qualification, method or decision, rather than judging them by general analytical reasoning.

It is known that lottery jackpot winners are highly publicized in newspapers, TV, and online media. This publicity contributes essentially to the availability heuristics, in the respect that the player focuses on the winner and ignores the millions of losers (Kahneman & Tversky, 1979), inducing themselves with the beliefs that “It could be me” and “I could be the winner in a future draw.” Besides, the jackpots could be really huge, and the amount itself potentiates the focus on winners (Phaneuf, 2024).

In particular, the magnitude factor contributes to explaining the 'against the odds' part of the question in the title, as chasing something assumes taking the risk and some “peripheral blindness” during target tracking. This principle also applies to jackpot chasing in slots.

Optimism bias, personal luck beliefs, and the illusion of control

Many players believe they are luckier than others, a form of optimism bias (Rogers & Webley, 2001). Many studies on lottery players have found that they often attribute wins to personal traits, lucky numbers, or fate, reinforcing the belief that they are “due” for a win (Ladouceur & Walker, 1996).

Other lottery players engage in superstitious rituals like choosing “lucky numbers” or purchasing tickets from specific locations (Dixon et al., 1998). This illusion of control makes them believe they can influence a purely random event like the lottery draw and increases their motivation to play (Wohl & Enzle, 2002).

Near miss effect

A near miss (for instance, matching three out of four numbers required for a win) can make players feel they are getting closer to winning, even though their odds remain unchanged. This effect, common in problem gambling, has been shown to be a reinforcer that encourages repeated play. The brain responds to near-misses in a way similar to actual wins, creating a sense of excitement that motivates continued participation (Brevers et al., 2015).

In the lottery, any two number combinations are independent as possible outcomes and have the same probability.

In probability terms, they are elementary events of the sample space with the same status. Combination (3, 17, 28, 40, 42) is not statistically related in any way to combination (3, 17, 28, 29, 50), even though they share common numbers. As such, there are no near-missing combinations, but just losing and winning combinations.

Regret aversion and fear of missing out

People are usually afraid of the regret of not taking an action that brought favorable results for others instead or could have been favorable for them. In the lottery, this regret aversion manifests for a player if they think that if they don't play for a draw, they might regret it if they see their usual numbers winning. As a result of this anticipated regret, the players continue buying tickets to avoid the emotional pain of a missed opportunity (Zeelenberg, 1999).

Another form of taking regret as a criterion for action is engaging in chasing previous losses (as an investment in lottery tickets bought), which obviously assumes constant play and sometimes spending more to “make up” for past losses. This behavior of chasing losses applies generally in gambling and is a risk factor for problematic gambling. In particular, for the lottery, many regular players argue that they cannot give up before recouping previous investments, a kind of social trap (Bennet, 2016).

Lottery as entertainment and aspirational spending

Some players view the lottery as a form of affordable entertainment rather than as an investment. Studies have found that most lottery players are seeking the thrill of the possibility of winning, fantasizing about excessive wealth; it's about spending happy time imagining “what if” (Anderson & Mitchell, 2018). A study found that lottery participation increased the happiness of participants before the draw, and thus, part of the utility of a lottery ticket is consumed before the draw (Burger et al., 2020).

The dream of a life-changing win provides excitement and hope, particularly among lower-income individuals who see the lottery as one of the few available paths to wealth (Beckert & Lutter, 2013).

Social and cultural influences

In many communities, playing the lottery is a social norm or habit. Friends, family, and co-workers often participate together, and players feel encouraged to continue buying tickets to stay included (Haisley et al., 2008). Besides, lotteries are sometimes advertised as supporting good causes (e.g., education or public projects), and that makes lottery players feel their participation is justified beyond just gambling.

Conclusion

The mathematical facts of the lottery are a bit hard to understand and interpret for those not inclined to math, but at least such information is transparent and retrievable; thus, anyone can come to be aware of the magnitude of the probabilities of winning the lottery.

The factors that may explain why lottery players persist in playing despite the minuscule chances of winning are mathematical, cognitive, psychological, and social. It is difficult if not impossible to determine which of these has the greatest weight in a general explanation of this fact, and research has not yet done so, the main reason being that decision-making behavior depends on the personal profile of each player.

The presence of the psychological factor of entertainment seems to be applicable to all profiles, which keeps the lottery in a relatively safe zone with respect to problem gambling, even if this game also has its own addictive potential.

Selling hope is, therefore different in casino-style than in lotteries, and lottery players can spend reasonably for fun and having dreams, regardless of whether they chase or not the big win.

Footnotes:

1. Statista. 2024. Share of population participating in gambling in the United States from 2021 to 2023 by type of gambling. Statista.com.
2. Gambling Commission. 2024. Statistics on gambling participation – Year 2 (2024), wave 1: Official statistics. Gambling Commission.
3. Yogonet Gaming News. 2024. Australia: Study shows decline in gambling participation, rise in online betting and risky behaviors. Yogonet Gaming News.
4. Barboianu, C. 2009. The mathematics of lottery: odds, combinations, systems. Infarom.
5. Barboianu, C. 2014. Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation. Journal of Gambling Issues, Vol. 29, 1-23.
6. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291.
7. Phaneuf, T. (2024). How the Lottery Works and How Much You'd Keep If You Won. Nerdwallet.
8. Rogers, P., & Webley, P. (2001). 'It could be us!': Cognitive and social psychological factors in UK National Lottery play. Applied Psychology, 50(1), 181-199.
9. Ladouceur, R., & Walker, M. (1996). A cognitive perspective on gambling. Cognitive and Behavioral Practice, 3(1), 58-63.
10. Dixon, M. R., Hayes, L. J., & Ebbs, R. E. (1998). Engagement in superstitious behavior by college students in a simulated gambling task. Psychological Reports, 83(3), 959–962.
11. Wohl, M. J. A., & Enzle, M. E. (2002). The deployment of personal luck: Illusory control in games of pure chance. Personality and Social Psychology Bulletin, 28(10), 1388-1397.
12. Brevers, D., Cleeremans, A., Goudriaan, A. E., Bechara, A., Kornreich, C., Verbanck, P., & Noël, X. (2015). Decision making under uncertainty and near-miss effect in pathological gamblers. Psychological Medicine, 45(12), 2661-2672.
13. Zeelenberg, M. (1999). Anticipated regret, expected feedback and behavioral decision making. Journal of Behavioral Decision Making, 12(2), 93-106.
14. Bennett K. (2016). 6 Reasons We Keep Playing the Lottery. Psychology Today.
15. Anderson, R. & Mitchell, D. (2018). Seven reasons we play lotto – even though we know we probably won't win the jackpot. The Conversation.
16. Burger, M.J., Hendriks, M., Pleeging, E. & van Ours, J. C. The joy of lottery play: evidence from a field experiment. Experimental Economics, Vol. 23, 1235–1256.
17. Beckert, J., & Lutter, M. (2013). Why the poor play the lottery: Sociological approaches to explaining class-based lottery play. Sociological Forum, 28(4), 1172-1196.
18. Haisley, E., Mostafa, R., & Loewenstein, G. (2008). Subjective relative income and lottery ticket purchases. Journal of Behavioral Decision Making, 21(3), 283-295.