What are your odds of winning in a lottery?
The mathematics of winning: how small amounts spark big dreams
Confession time: For the past several years, I’ve been buying lottery tickets. Not one, but two, if not more, each month. It's driven by hopes for a rare life-changing win.
I’m part of a joint “enterprise” among workmates, a sort of in-office “syndication”.
We pitch in small amounts to join the draw together. This is quite common. And it can happen that small amounts could go big – if we do win. And that's exactly what lurks in every ticket-holder's mind.
There's a self-assured line we take comfort in: “So, if that person won, surely I (or we) will win, too.”
There's a shared sort of thrill. Does each story you read about a lottery winner raise your “odds” of winning? It seems it does.
Attention grabber
It turns out every story about a winner grabs attention like no other: a life-changing story, driven by rarity and “me-too” type of what psychologists call "hope mechanism".
The way lotteries are presented (e.g., highlighting jackpots) influences participants' perceptions, making them more willing to play despite low odds
And it’s true with every person, in every country, according to studies.
Every story about a lottery winner draws more eyeballs than any apocalyptic news of space debris or a catastrophic meteor strike threatening Earth.
Hope mechanism
It’s not just the thrill — it’s the psychology.
Researchers, including from Carnegie Mellon University, reveal that lotteries offer a hope mechanism. They serve as an escape, especially for those facing financial struggles.
Our real odds of winning? The probability of winning? It's rather low.
Yet the act provides optimism. There’s a fleeting sense of control. Lotteries promise not just riches but the dream of “upward mobility,” however improbable the odds.
The mathematics of winning
Mathematically, the odds of winning a lottery depend on the number of possible combinations.
This is determined by the range of digits and the specific rules of the lottery.
Here's an example calculation:
4-digit number:
With 10 possible digits (0–9) for each position, the total combinations are 10 to the 4th power = 10,000,
10^4 = 10,000; Odds: 1 in 10,000
5-digit number:
10^5 = 100,000; Odds: 1 in 100,000
6-digit number:
10^6 = 1,000,000; Odds: 1 in 1,000,000
7-digit number:
10^7 = 10,000,000 Odds: 1 in 10,000,000
8-digit number:
10^8 = 100,000,000; Odds: 1 in 100,000,000
For lotteries with additional conditions (e.g., matching specific orders or subsets), the odds would be lower. (Note: ^ means raised to nth power)
Psychology
There are actually a number of studies examining the psychology behind ticket draws:
#1. Risk and reward perception in lotteries:
A landmark 1979 study by Daniel Kahneman and Amos Tversky explored how people overweight small probabilities, making lotteries appealing despite their low odds of winning. The research showed that the excitement of a potential reward outweighs rational risk assessments for many individuals.
The study ("Prospect Theory: An Analysis of Decision under Risk", now published in the digital library JSTOR) outlines the risk and reward perception in lotteries include offer the following key points:
- a) Overweighting small probabilities: People disproportionately value low-probability outcomes, making the idea of winning a lottery seem more likely than it is.
- b) Emotion-driven decision making: The excitement and emotional appeal of potential rewards often override logical assessments of risk.
- c) Framing effects: The way lotteries are presented (e.g., highlighting jackpots) influences participants' perceptions, making them more willing to play despite low odds.
#2. Lotteries as a “hope mechanism”
In a 2008 research, Emily Haisley at the Tepper School of Business Carnegie Mellon University stated that lotteries offer a hope mechanism.
She demonstrated that lotteries serve as a psychological escape, giving participants a sense of control and hope, especially in financially constrained situations.
She emphasised that participation in lotteries provides psychological benefits — such as a sense of optimism and control, particularly for individuals in financially constrained situations.
Her work suggests that lotteries act as a form of escapism, offering participants a brief respite from financial pressures and fostering a belief in “upward mobility” — even if the actual odds of winning are minimal.
#3. Illusion of control
The “illusion of control”, coined by Langer and Roth (1975), describes how people overestimate their ability to influence random events, enhancing engagement in tasks like lotteries. Their study showed that early successes in a chance task (predicting coin tosses) fostered a “skill” mindset.
Participants misremembered wins and expected more future successes. This cognitive bias also explains overconfidence in broader contexts, such as health systems, where unintended consequences can undermine perceived control.
Einstein encapsulated this with: “We cannot solve problems with the same thinking that created them”.