Originally published at: https://boingboing.net/2020/12/07/how-a-convience-store-owner-hacked-the-lottery-odds-and-won-27-million.html

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There was a 60 Minutes segment on this last year as well.

I must be missing something in the math: In the rolldown weeks, the 3-match was worth $50 on a 1-in-54 chance, and the 4-match $1000 in a 1-in-1500 chance. How are the odds in the player’s favor?

It seems to me the house still always wins (unless you’re a Trump-owned casino).

I like stories like this, where smart and persistent people who aren’t morally and ethically challenged (e.g. who aren’t Republican politicians) take the time to figure out how to game a broken system with math.

This fellow reminds me of the “Press Your Luck” hackers, Michael Larson.

For anyone who’s prone to criticise the Selbees, this quote needs to be pulled:

Meanwhile, around them, the larger American economy was imploding. The housing bubble, the bank bailouts, the executive bonus scandals, the automotive bankruptcies—panic, panic, panic, panic. In Evart, an auto glass plant that had supplied Chrysler closed down, throwing 120 people out of work. American corporations had been playing a lot of games, noted Jerry, and their ways had finally caught up. “They were taking far more risks than I was, based on their rewards. That’s why I did a risk-reward analysis after every game, to make sure I was still on track.”

Compared with Bear Stearns or Goldman Sachs, the Selbees were downright conservative. By 2009 they had grossed more than $20 million in winning tickets—a net profit of $5 million after expenses and taxes—but their lifestyle didn’t change.

Also, when lottery officials were asked about the flaws by a reporter…

“It was pretty obvious that something was askew,” Estes said. She requested public records from the lottery and discovered that other groups had formed to buy tickets, including one with a bunch of MIT students. When Estes asked officials for comment, however, they claimed ignorance. “The lottery was really sleazy about the whole thing,” she said. “They were quite aware this was going on, and they acted shocked when I told them about it.”

Probably because the odds are aggregate. For one $1 ticket, you are buying into odds for matching two, three, four, five, or six numbers. So the expected value of the ticket is the sum of the expected value of each payout for a number match.

For instance, there are some jackpot lotteries that have an expected value ratio above 1.0 just from the jackpot when the jackpot grows every draw. That doesn’t mean you’re more likely to win, it just means that the payoff times the odds is worth more than the buy-in.

right right right - each “pick 6” is actually 20 (if I’ve got my combinatorics right) sets of 3.

No, that’s usually included in the published odds.

What it means is that the expected value of the $1 bet is higher than $1, because the aggregate of the paths to payout sums up to more. So, for example, (expected value of matching 2 numbers is 0.2 x $1) + (expected value of matching 3 numbers is 0.0185 x $50) + (expected value of matching 4 numbers is 0.00067 x $1000) + (expected value of matching 5 is 0.00005 x $15000) + (expected value of matching 6 is 0.0000002 x $4M). That sums up to a really high expected value, $3.34, so probably not realistic, but that’s the kind of hack we’re talking about here and why it’s legal. They aren’t hacking the lottery system, they’re merely timing the investment of large sums to when the expected value reaches a maximum value.

The values of the 2-5 matches are contingent on there being no 6 match, so you need to subtract the $0.8 from the possible 6 match and scale down slightly to account for the chance someone else gets all 6. ~~Also the odds need to be adjusted because you can’t double count (if you get 3 on a ticket, you can’t also use it for a 2 payout).~~ It still seems to come out to a crazy high expected value of just over $2 per ticket, though. (since it depends on whether someone else matches all 6, the value goes down the more tickets other people buy)

eta: edited based on Duke’s later post

Ah, thanks. I forgot that the 6 match win blocked the prize multiplier for this lottery.

I get that one ticket can’t win multiple prizes, but I thought that was negligible because it only represents one alternate outcome per match level?

In a perfect world this would be a great advertisement for learning math, promoted by the public sector to encourage students. Maybe it would even increase the compliance rate for the tax on people who are bad at math. Oh wait, those are the people who fund math education. Hmm. Guess it’s going to stay bad math all the way down.

Apologies, I’m a bit time-poor today - does the article mention whether this hack even dented the Lottery Commission’s bottom line? Or was the turnover and profit high enough that nobody even noticed?

This suggests a system where the people operating the lottery only have revenue-based incentives, and no disincentive for overall losses. Their incentive was to let this broken game run for as long as possible, with as many players as possible aware that it was broken. It’s the same thing that gave us the mortgage crisis in 2006-2010.

I love to hear about situations like this, where people are essentially exploiting a poorly configured system to print money. But I hate that these lotteries pretend to operate for the public good. What a bunch of scumbags.

You may be right, I didn’t check the details on the game to calculate the odds myself. If the stated odds of a 20% chance for a match 2 is for exactly 2 (not at least 2) then no adjustment needs to be made. If the odds are for 2 or more (and ≥ 3, etc) then it would be a ~10% reduction in the sum value.

That’s correct. The odds for any particular match (and thus prize) is for just that match, excluding the ones that are greater.

I’m glad that they do those calculations for me. I do a lot of stats for work, just not that kind.

Roll downs and roll overs are mechanisms explicitly designed to make sure that ~50% of the money taken in is returned to players in the form of prizes. They are there so that the lotteries aren’t TOO profitable. Lotteries are incredibly profitable, and they don’t care whether that “limiting mechanism” goes to benefit a few players or many. But once EVERYBODY realized, they probably had to stop the “roll down” mechanism.

Certainly when the powerball or mega millions prize gets really big, it would theoretically be profitable to buy every single possible ticket if you didn’t split the jackpot.* The jackpots grow so big in part because the “random pick” function is imperfectly random. The algorithm is more likely to pick certain combinations of numbers than others. That has no bearing on the probability of those numbers winning, but it DOES impact the probability of splitting the prize. Since they sell more tickets when the jackpot is big, but the jackpot doesn’t get multiplied by the number of winners, an imperfect system of random picks for suits the lotteries. Heads, nobody wins the jackpot, it grows and they sell more tickets for the next drawing. Tails, and they only pay out the jackpot amount split between the winners (plus the non jackpot prizes).

Edited to add: Since the maximum jackpot payoff for a particular drawing is fixed, There is no real risk that large numbers of tickets being sold will make an individual drawing unprofitable for the PB and MM lotteries. But since the non-jackpot prizes were being increased in this case, if too many people bought tickets the losses could be great.

*they will not sell tickets this way.

Thanks, that clears up a few things. If I’ve understood you correctly: no impact on profit (because lotteries), but poor optics about it going to a few people over and over, plus “if everyone does this …”

My probability prof told us that if we want to gamble, to please go to a casino. Compared to lotteries, they’re giving money away.

In the UK the lottery jackpot keeps getting bigger each time (I think it’s now drawn twice a week) - I vaguely remember seeing a documentary about a group that tries to buy every combination, for a theoretical modest profit as long as the jackpot wasn’t shared - I can’t find anything online about it though. I think they had a problem as someone refused to accept their pre-printed slips late in the day; they got the money in the end but we’re on the hook for a massive loss if it went wrong. (It could be I’m misremembering)

A few years ago they increased the number of balls from 49 to 59, probably to stop this

I had a conversation at work with somebody that said “The worst thing about buying a lottery ticked is all the time you waste thinking about it.” To which my response was “In my mind, all that time thinking about it IS the payoff. To me, buying a ticket is paying rent on my fantasy. It’s not like you are going to win.” I often buy single tickets when the jackpots are big.

I’m the same way. I’ll buy a single ticket now and then only if the jackpot is at least 10-million times the ticket price and then enjoy the fantasy for a bit. I don’t really think I’ll win, though, and as someone who understands how probability works you likely don’t either.

The person you were talking to may have been buying more tickets more frequently and then actually thinking about whether or not he’ll win. It’s closer to a habitual gambler’s mentality, which isn’t pleasant.

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