Distortions in Lottery and Horse Racing Odds and Expected Value Bugs: Verifying Everything from LOTO Rolldowns to toto Match Cancellations and Year-End Jumbo Fraud Allegations
Chapter 1: Is the Winning Probability of the Year-End Jumbo Lottery Unmanipulatable? The Mechanism of the Unit System and the Lucky Store Effect
Conclusion: Bias in sales locations does not affect probability / The only strategy is a minimum recovery guarantee through the control of 'variance'
Variables in this chapter (keys to the strategy)
Things that can be manipulated: Variance (risk avoidance), guarantee of minimum recovery rate
Things that cannot be manipulated: Drawing probability (the realm of God), upper limit of expected value
What do we call 'probability'? Physical probability, expected value, and structural distortion
'Lottery ticket counters do not have all the numbers. Therefore, isn't the winning probability different depending on the counter?'
This simple question of yours is a very sharp inquiry in probability theory.
We usually use the term 'winning probability' as a catch-all, but it actually contains three different layers that should be separated.
Physical Probability (Chance): The roll of the dice determined by God.
Expected Value (Expectation): The size of the pie that humans share.
Structural Bias (Structural Bias): The gaps created by rules and crowd psychology.
Tonight, we will trace the 'true nature of probability' inherent in each form of gambling, from Jumbo lotteries to LOTO, toto/BIG, and horse racing, as well as the records of the madness of researchers who challenged those solid walls and people who sometimes attempted 'strategies' that defied common sense.
First, let's start with the world of the 'Year-End Jumbo Lottery,' which is the most familiar yet the most difficult to strategize.
Explanation of the 'Unit System' mechanism: Why do lines form at the Nishi-Ginza Chance Center?
Why do long lines form every year at 'Window No. 1 of the Nishi-Ginza Chance Center'?
Many people believe that 'it's because a lot of winning tickets are distributed there.'
However, if you understand the 'unit system,' which is the operating system for lotteries, you will realize that this belief has nothing to do with 'probability.'
What is the '20 million ticket set' structure of the Jumbo Lottery?
Jumbo lottery tickets are not printed in a disorderly fashion.
For example, in the Year-End Jumbo Lottery, there is a strict set known as “1 unit = 20 million tickets”.
Composition: Series 01, No. 100000 to Series 200, No. 199999
Number of 1st prizes: Within this set of 20 million tickets, there is always one 1st prize (equivalent to three prizes including the front and back prizes).
These “20 million ticket sets” are distributed across the country in dozens of sets (units) depending on the total sales volume.
Certainly, physical “biases” do exist. A small ticket booth in a rural area might only receive “Series 01 to 10 of Unit 01.” A large store like the one in Nishi-Ginza might receive “all series from Units 01 to 50.”
Since the winning tickets are determined by a “public drawing after sales end,” the place of purchase is irrelevant to the probability.
However, this inventory bias does not move your winning probability by even a millimeter.
This is because the lottery drawing is held at a public event “after sales have closed.”
If this were a “scratch-off lottery,” it would be a different story. Since scratch-offs are determined the moment they are printed, there is meaning in looking for a store that still has winning tickets left.
However, the Jumbo Lottery is different. The “winning number” is only born into this world the moment the drawing machine (wind-wheel board) spins and the arrow lands.
Even if the number you bought was one that only had a single ticket delivered to that store, if the drawing machine selects that number, it is a winner. Conversely, even if a number was delivered in large quantities, if it is not selected, it is just a piece of paper.
“Which store you buy from” is not a question of “which river you cast your fishing line into.” The fish (the win) are randomly released by God (the drawing machine) after you have cast your line. Therefore, the probability of $${1/20,000,000}$$ remains constant no matter where you are in Japan.
The true nature of “lucky stores” is the Lucky Store Effect—an explanation via behavioral economics
Even so, if you look at the data, “stores where high-value wins frequently occur” do exist. Is this occult? No, this is a phenomenon that can be explained by behavioral economics research called the “Lucky Store Effect.”
Research by Guryan and Kearney (2008) published in the American Economic Review analyzed lottery sales data in Texas and demonstrated the following cycle.
Occurrence of chance: A high-value win occurs by pure chance at a certain store.
Diffusion of information: Information that “a millionaire was born from that store” spreads.
Rush of the crowd: Customers seeking good luck flock to the store, and the store's sales volume (denominator $${N}$$) increases rapidly.
Increase in frequency: Even if the probability $${P}$$ does not change, if the number of trials $${N}$$ increases, the number of wins $${E = N \times P}$$ increases.
Reinforcement of belief: The reputation that “that place really is lucky!” becomes established, and the lines grow even longer.
In other words, it is not that "people line up because they win," but rather “people win (in terms of frequency) because they line up.”
This phenomenon is a classic example of a "self-fulfilling prophecy." The act of standing in line is meaningless as a ritual to increase probability. Even so, if you consider it a cost for the excitement of participating in a festival, it might not be a waste at all.
The only strategy for Jumbo Lottery is risk management (control of variance) through "set buying."
While you cannot intervene in the drawing probability itself in the Jumbo Lottery, there is one variable that players can manipulate. That is "variance (risk)."
By using the mechanism of "set buying," you can slightly change the shape of the probability distribution.
The difference between "sequential numbers" and "assorted numbers"—certainty or excitement?
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Sequential (10 tickets): A set where the group and the first five digits are the same, and the last digits are 0–9.
Effect: The winning probability for the lowest prize (e.g., 300 yen) becomes 100%. Also, since you can aim for the 1st prize and the surrounding prizes simultaneously, the explosive power (maximum value) when you win is higher.
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Assorted (10 tickets): A set where the numbers are not sequential, but the last digits 0–9 are all present.
Effect: The lowest prize is also secured here, but there is a risk of missing the surrounding prizes when winning the 1st prize. Instead, because the groups and numbers are scattered, it has the effect of reducing the feeling of "not even coming close" (distributing the excitement).
Pursuing guaranteed returns with "Fukuren 100 / Fukubara 100"
As an even more obsessive way to buy, there are "Fukuren 100" and "Fukubara 100" options supported by some sales outlets.
This is a way of buying where you purchase a 100-ticket set (30,000 yen worth) and ensure all last two digits (00–99) are covered.
Guaranteed return: Because you cover all last two digits, you are guaranteed one win for the last two digits (several thousand yen) and ten wins for the lowest prize (300 yen).
Meaning: This ensures that about 20% to 30% of the purchase amount is guaranteed to be returned (*depending on the prize structure of the specific draw).
This is not a "strategy to make it easier to win." It is merely an act of paying 30,000 yen to reliably get back about 6,000 to 9,000 yen.
However, this reveals the desperate human desire to "reduce uncertainty as much as possible." The greatest "strategy" in the Jumbo Lottery is to stop chasing further, minimize losses through guaranteed recovery, and keep dreaming—this is the essence of sound asset management (?).
Digression: The "dark side" and "fraud" of the lottery—is cheating even necessary in a collection system with a house edge of up to 54%?
"The lottery is a malicious form of gambling that even the Yakuza wouldn't touch."
Conversations on social media, at bars, and in hair salons. A bottomless distrust of lottery management swirls there.
"Where do the 10 billion yen in unclaimed winnings that appear every year disappear to? Into the management's pockets?" "That analog bow and arrow could easily be manipulated with magnets." "Why does only Mizuho Bank have a monopoly? The management and oversight are too close."
These grievances are justified.
In Chapter 1, I stated that "lotteries are a matter of luck," but since they are operated as a system, it would be dishonest from an engineering perspective to assert that there is zero room for “exploits” to enter.
Therefore, I will digress from the main argument and, instead of dismissing these "suspicions" as mere conspiracy theories, I will conduct a thorough simulation from the perspective of a malicious auditor: “If the operators were truly corrupt, how would they pull it off?”
“The 54% House Edge” is the Ultimate Defensive Wall
First, regarding the suspicion that "the operators are cheating to skim money." To put it bluntly, the greatest irony is that “they have no motive to cheat.”
The payout ratio for lotteries is approximately 54%. For every 10,000 yen sold, 5,400 yen is immediately secured as the operator's share. (While the figures fluctuate, by law, the source of prize money is capped at a maximum of 50% of total sales because "the total amount of prize money cannot exceed 50% of total sales." The remainder goes toward profits and expenses.)
This is the important part. Why would they risk prison to "manipulate arrows" or "tamper with numbers" to skim small change when they already possess a system that generates massive profits effortlessly as long as they follow the rules?
There is no business owner who would go out of their way to engage in "illegal collection" at the risk of imprisonment when a "legal collection system" is already complete. For them, the only real fear is that the fraud will be exposed and this "lucrative collection system" will be shut down.
The Cost-Performance of “Arrows” and “Magnets”
The persistent suspicion that "the lottery arrows are manipulated by remote controls or magnets." If you were the mastermind of an evil organization, would you actually do this?
Requirements: A custom-made pinwheel board, remote control devices, and the bribery of the installation contractors, the operators on the day, the supervising witnesses, and the entire broadcast staff.
Risk: If even one person involved leaks it to a "weekly magazine," the organization collapses.
The same applies to number-selection lotteries like Loto. There is a suspicion that "numbers are chosen after the deadline," but the draws are broadcast live online.
Considering the cost and risk of staging "movements that defy the laws of physics" in front of the public, “simply generating results randomly and skimming the 54% fee” is far cheaper and more reliable.
The True “Darkness” is Not Malice, but “Monopoly and Negligence”
So, is the Japanese lottery a clean and honest system?
The fact that we cannot assert this is where the problem runs deep. The lax governance, such as “concentration of operations at Mizuho Bank” and “amakudari (descent from heaven) structures,” are clear security risk factors (single points of failure).
Actual “Configuration Errors” That Have Occurred
Even before malicious fraud, simple mistakes have occurred. In 2024, regarding an internet-only lottery (Quick One), a sales suspension and refund incident occurred because “due to a configuration error, a different number of winning tickets were allocated than what was announced at the time of sale.”
This was not a case of "someone manipulating things behind the scenes." It was an incident that exposed the “sloppiness of the operational structure,” where an organization responsible for a massive system made a simple configuration error that went unchecked until just before sales.
What is scarier than "conspiracy theories" is this kind of “incompetence and negligence” typical of such massive organizations.
If they had the ability to concentrate winning tickets at specific sales outlets or highly manipulate the lottery balls, they wouldn't be making such elementary setup mistakes in the first place.
Conclusion: There is no need to cheat
I do not intend to deny the voices of suspicion. However, as I have repeated many times,
“This system is designed so that the house always wins, without the need for any rigging.”
Regardless of how unclaimed prize money is handled or whether there is a bias in sales outlets, by agreeing to the rules of taking more than half as the house edge when buying a ticket, we are consenting to being “exploited.”
If you have the energy to suspect fraud, it might be more constructive to use that anger as fuel to search for the “odds distortions (market bugs)” explained in Chapter 4.
Ultimately, the biggest “cheat” is not some illegal act performed in secret by someone, but the rule of “a payout rate of less than 50%” itself,which is being carried out openly in broad daylight.
Chapter 2: The Strategy for Loto 6 and Loto 7 is Maximizing “Expected Value”—Avoiding Bias and Aiming for Carryovers
Conclusion: You cannot change the winning probability, but you can manipulate the probability of “winning it all” by choosing unpopular numbers
Variables of this chapter (keys to the strategy)
Things you can manipulate: Expected value (your share), avoiding the risk of splitting the pot
Things you cannot manipulate: Drawing probability. Enemy: Human psychological bias (birthday buying, pattern buying)
The enemy of Loto is not the “drawing machine,” but “people who buy the same numbers”
Once you enter the world of “LOTO” or “Numbers,” where players choose their own numbers, the rules of the game change completely.
Here, the variable you should be moving is not the “probability of winning.” It is“the share you get when you win (expected value).”
The Loto drawing itself is physically performed by a dedicated drawing machine (such as the Yume Loto-kun), so there is no room for intervention here, just like with the Jumbo. However, the prize money is determined by the “pari-mutuel system (splitting the sales among the winners).”
In other words,“not buying the same numbers as others”is the only and strongest strategy to maximize expected value.
Hacking human psychological bias—avoiding birthdays and pattern buying
Humans are surprisingly bad at choosing numbers randomly. Numerous behavioral economics studies (Judgment and Decision Making, 2016, etc.) have revealed strong biases among lottery buyers. Avoiding these is the first step to a winning strategy.
The risk of splitting the pot due to “birthday bias”
This is the most famous bias. Many people choose their own or their family's birthdays (1st to 31st).
Combinations consisting only of numbers from 1 to 31 have an overwhelmingly higher number of buyers compared to combinations that include numbers from 32 to 43 (in the case of Loto 6). If the first prize is won with numbers like '01, 05, 10, 15, 20, 25', the number of winners could reach hundreds, and the prize money could plummet to a fraction of the theoretical value (not tens of millions, but mere millions of yen).
The Trap of 'Design Buying' (Pattern Buying)
There is a certain segment of people who choose numbers to form 'straight diagonal lines,' 'crosses,' or 'heart shapes' on the mark sheet. This also maximizes the risk of having to split the prize if you win.
'Gambler's Fallacy' and 'Hot Numbers'
The psychology of avoiding numbers because 'the same numbers from the previous draw won't come up' (Gambler's Fallacy), or conversely, the psychology of chasing numbers because 'these numbers have been coming up a lot lately' (Hot Numbers) also comes into play. While these are mathematically meaningless guesses, they work powerfully as crowd psychology.
In other words, to outsmart your rivals, you should intentionally choose 'large numbers of 32 or higher' or 'irregular and aesthetically unpleasing sequences'. This does not change the probability of winning the first prize, but it dramatically improves your 'Solo-Win Probability'.
Is 'Buying Every Combination' Possible? — Stefan Mandel and the Virginia Case
'If you can't manipulate the probability, you should just buy every combination.'
There are madmen who did not let this diabolical idea end as a mere theoretical exercise, but executed it as a real-world 'logistics project'.
Expected Value Calculation and Execution by Mathematician Stefan Mandel (1992)
Romanian-born mathematician Stefan Mandel calculated that if you target a 'lottery where the carryover has accumulated sufficiently and the total number of combinations is relatively small,' the total prize money will exceed the cost of purchasing every combination (the expected value becomes positive).
He set his sights on the Virginia state lottery (at the time, there were about 7 million possible combinations).
However, the real enemy here is not mathematics. It is physical time and terminal operations.
Challenge: Within the few days before the draw, you must print 7 million mark sheets and have them read by the terminals at the sales outlets.
Strategy: He formed an international investment fund and mobilized dozens of printers and hundreds of personnel. They occupied terminals at supermarkets and gas stations and proceeded with the purchases in an organized manner.
As a result, although he failed to purchase every combination due to the deadline (he purchased about 70%), he was lucky enough to have the winning ticket among them, and he successfully hit the jackpot. This was not 'gambling,' but a brute-force effort closer to large-scale 'civil engineering'.
The Massachusetts 'Cash WinFall' Incident that Exploited the Rolldown Method
A more sophisticated case of an 'expected value hack' occurred in Massachusetts in the late 2000s.
The 'Rolldown (Prize Distribution)' Bug when the Cap is Reached
The lottery known as "Cash WinFall" had a special rule.
It stated that "if the carryover reaches the cap and there is no first-prize winner, the excess prize money is not carried over but instead distributed to lower prize tiers (3rd or 4th prize) (rolldown)."
Normally, winning the first prize is extremely difficult, but the odds for 3rd or 4th prize are realistic. When a rolldown occurs, the payouts for these lower tiers jump to 5 to 10 times their usual amounts.
Calculations revealed that only during the week a rolldown occurs, the expected value per ticket clearly exceeds the purchase price (a state where you buy for $1 and get $1.20 back).
Systematic exploitation by a group of MIT students
It was multiple groups, including a group of MIT (Massachusetts Institute of Technology) students and former biomedical researchers, who noticed this bug.
They accurately predicted the "weeks when rolldowns would occur" and invested funds on the scale of tens of millions of yen at those moments.
Rather than aiming for the first prize like Stefan Mandel, this method of reliably scooping up the inflated lower-tier prize money was an "investment" that minimized risk to the extreme. It is said that they earned millions of dollars in profits over the several years before state authorities changed the rules and abolished the lottery.
You cannot beat the god of probability, but you can beat the loopholes created by humans.
Chapter 3: A Winning Strategy for toto/BIG? The "Probability Bug" Created by Match Cancellations and Combinatorial Mathematics
Conclusion: MEGA BIG's probability fluctuates by 256 times / Minimum ticket guarantee using the Football Pool Problem
Variables in this chapter (keys to the strategy)
Things that can be manipulated: Rule application (use of conditional probability)
Things that cannot be manipulated: Match results, random generation (in the case of BIG) Singularity: "Deemed winning" due to match cancellation
What are the "loopholes" hidden in sports lotteries (toto/BIG)?
Japan's sports promotion lotteries (toto/BIG) have unique "specifications" not found in other lotteries. This concerns how they handle situations when "match cancellations" occur due to weather disasters or similar events.
Normally, the probability of winning a lottery is mathematically fixed, but in toto/BIG, there are moments when the probability space itself is distorted due to external factors (typhoons or heavy snow). Peering into this "rift" is the theme of this chapter.
The 1476th draw where MEGA BIG's probability became 256 times higher due to a typhoon-related match cancellation
In the summer of 2024, when a typhoon hit Japan, a "probability fluctuation" rare in lottery history occurred. Yes, it was the legendary 1476th MEGA BIG draw.
Cancelled matches are treated as "matching all outcomes"
MEGA BIG is a lottery where the total scores of 12 designated matches are randomly assigned from four choices: '1, 2, 3, 4'. The theoretical probability of winning the first prize is $${1/4^{12}}$$, which is an extremely difficult challenge of about 1 in 16.77 million.
However, in this round, out of the 12 target matches,4 matches were cancelled occurred.
Under the official rules, cancelled matches are considered 'a win (match) regardless of whether the number is [1][2][3][4]'.
In other words, it was no longer necessary to guess all 12 matches correctly; effectively, one only needed to guess 8 matches correctly.
The calculation is simple and dramatic.
$$
4^4 = 256
$$
The probability of winning jumped to256 times the theoretical probability, or about 1 in 65,000.
The Trap of Pool Payouts—Prize Money Plummeting Due to a Rush of Winners
People with high mathematical literacy who noticed this 'bug,' as well as those who heard about it on social media, rushed to the ticket counters.
'This is a win'—that intuition was correct from a probabilistic standpoint.
However, the outcome was ironic.
Because the probability increased by 256 times, naturally,the number of winners also exploded. While there are usually 0 to 1 first-prize winners, this round saw a staggering269 winners emerge.
BIG prize money is 'distributed from sales (pool system),' and there is also a cap on the carryover. As a result, the limited funds had to be split among 269 people, and the first-prize money fell far short of the 1.2 billion yen cap, remaining at approximately 24.8 million yen.
Even so, it is analyzed that there was a high possibility that the 'expected value was positive (if you bought 100 yen worth, you would theoretically get back more than that),' including second-prize wins and below. This was a modern, alchemical moment created by weather conditions and rules.
The Unsolved Problem in Combinatorics: 'Football Pool Problem' and Covering Code
While BIG is chosen randomly by a computer, 'toto,' where you make your own predictions, has a profound problem that mathematicians have been tackling for a long time.
That isthe 'Football Pool Problem'.
A Mathematical Approach to Creating a 'Third-Prize Guarantee' with the Minimum Number of Tickets
If you buy all possible combinations for toto (13 matches, 3 choices of win/loss/draw), it comes to about 1.6 million combinations ($${3^{13}}$$ ). This is unrealistic for an individual.
So, then,'What is the minimum number of tickets one must buy to ensure at least one third-prize win?'
This question goes beyond simple gambling strategy and becomes a problem of advanced combinatorial mathematics known as "Covering Code".
For example, a problem like "What is the minimum set of combinations to ensure at least 4 out of 5 match results (3-way choice) are correct?" actually has aspects of an "unsolved problem" where, for some $${n}$$, a complete solution (the minimum number of tickets) has not yet been determined.
Researchers continue to use supercomputers to search for combinations in order to update this minimum number (the upper bound).
For them, toto is no longer gambling. It is the frontier of human knowledge, applied to error correction technology in communications and the analysis of genetic sequences.
Information is power—those who master public information grasp the fissures in probability.
What the world of toto/BIG teaches us is the truth that "Information is power."
Typhoon path forecasts, match cancellation rules, and the accumulation status of carryovers. Only those who correctly process this publicly available information and take action can reach into the "fissures in probability" that open for only a brief moment.
While you cannot change God's dice (match results or random generation) themselves, discerning the stage on which the dice are rolled (the rule environment) is one of the few privileges granted to humans.
Chapter 4: The Winning Method for Horse Racing Lies Within the "Crowd"—Odds Distortions and the Wall of Liquidity
The illusion of "Lock," which pierces through the wall of the deduction rate to exploit odds distortions
Variables of this chapter (keys to the strategy)
Things that can be manipulated: Market price (odds), arbitrage
Enemies: Deduction rate (JRA's house edge), liquidity (your own shadow)
The enemy is not the organizer but "other purchasers"—the cruel truth of the pari-mutuel system
Finally, we step into the world of horse racing, the most human and most cruel of public sports. If lotteries and LOTO are a battle against "God (probability)," horse racing is a battle against "Humans (the crowd)."
Horse racing uses the "pari-mutuel system." This is a system where the money invested by participants is pooled, and after the organizer's share (deduction rate) is removed, the remainder is split among the winners.
The first obstacle that stands in the way is this wall of the deduction rate.
Looking at the payout rates published by the JRA (refer to business reports from 2025, etc.), you can see the height of that wall.
Win/Place: 80.0%
Quinella/Bracket Quinella: 77.5%
Trifecta: 72.5%
WIN5: 70.0%
The more you buy at random, the more your funds will surely melt away at this ratio. Compared to casino roulette (with a house edge of about 2.7% to 5.3%), this is an extremely harsh game.
However, why do "pros" and "syndicates" exist who overcome this high wall and continue to make a profit?
1. Odds distorted by "Longshot Bias"—what pros are looking at is not the horses, but the crowd's misconceptions
The odds (payout rates) for horse racing are not determined by the JRA, but by "the bets of other purchasers." And humans are not rational investors. This is where the opportunity to exploit arises.
A phenomenon known as "Favorite-Longshot Bias" has long been observed in the horse racing market.
Overvaluation of Longshots: People dream of a "come-from-behind victory" and tend to buy longshot horses with extremely low winning probabilities for more than they are actually worth. As a result, the odds for longshots become lower (less profitable) than the fair value.
Undervaluation of Favorites: Conversely, solid favorite horses are often avoided because the "payout is boring," and their odds tend to be higher (more profitable) than their actual ability.
Research by Snowberg and Wolfers (NBER, 2010) and others discusses whether this is due to "risk-loving (romantics)" or "probability misperception," but the conclusion that "odds do not accurately reflect ability" remains unshaken.
Professional horse racing bettors are not looking at the horses. They are looking at "distortions in the odds." The system of calmly continuing to buy "undervalued horses" that the crowd has ignored is what they are truly about.
2. The ultimate holy grail: "Lock (Arbitrage)"—the theoretical winning method and its limits
When the madness reaches its peak, they no longer even try to predict "which horse will win."
What they aim for is "Lock," known in financial terms as arbitrage.
Mechanism: If you exploit the contradiction between win bets and exacta bets, you can make a profit regardless of which horse wins.
In Japanese horse racing, there are multiple betting types for the same race, such as win, exacta, and trifecta.
In theory, these odds should be consistent. However, due to biases in voting and timing discrepancies, contradictions can occur for a split second.
For example, when comparing "win odds" with the "synthetic odds of an exacta total-coverage bet," if there is a distortion, it may be possible to allocate funds so that "no matter which horse wins, the payout will always exceed the purchase amount."
Evidence from the Japanese market: The severe reality of "Lock" occurring only twice in 175 races.
It sounds like a dream, but the reality is harsh.
Researchers at Kobe University (Ashiya, 2015) and others have verified the existence of this "Lock" using actual Japanese horse racing data.
Result: Out of 175 races, the conditions were met (a state where profit could be made without risk) in only 2 races.
Furthermore, there is a fatal paradox here.
The "liquidity trap." The moment you bet a large amount thinking "I can win!," your own bets lower the odds, causing the profit to vanish.
Unlike lotteries, in horse racing, the act of "you buying" itself changes the market price.
Just as "you cannot step on your own shadow," market distortions are destined to be erased by the hands of those who find them the moment they are discovered.
Beyond the wall of probability lies the "absence of a certain future" and the "value of delusion."
The "unmoving probability" of Jumbo lotteries, the "hidden side of human psychology" in LOTO, the "cracks in the rules" of toto, and the "market distortions" of horse racing.
What becomes visible at the end of the journey through these is the "there is no such thing as a certain future anywhere"—a boring and obvious fact.
Even for the Texas group that bought every possible combination, if someone else had happened to buy the same numbers, the prize money would have been halved, resulting in a massive loss.
The students of Cash WinFall also lost their "Holy Grail" due to rule changes by the authorities.
However, people still roll the dice.
Ultimately, the price we pay for gambling might not be for "increasing our money" itself, but for the right to indulge in the happy delusion of "what if I win" for the few days until the drawing or the few minutes while the race is running.
Because that is a cheaper and more certain "return" than any advanced mathematics or algorithm.