Impossible or improbable lottery results

There was a government-run lottery in the Philippines with a $4 million jackpot, and two improbable things happened. First, the numbers selected were all multiples of nine: 9, 45, 36, 27, 18, and 54. Second, 433 people won. The natural reaction from the public was that something sketchy happened, especially since the government has a history of sketchiness.

However, as statisticians and mathematicians do when rare and improbable events occur, they setup hypotheses and calculate probabilities. Terence Tao calculated the odds and noted that the lottery outcome was a highly improbable event under certain assumptions. But:

So this clearly demands some sort of explanation. But in actuality, many purchasers of lottery tickets do not select their numbers completely randomly; they often have some “lucky” numbers (e.g., based on birthdays or other personally significant dates) that they prefer to use, or choose numbers according to a simple pattern rather than go to the trouble of trying to make them truly random.

Nine happens to be a lucky number in some cultures. Also, as Tao notes, the multiples of nine form a diagonal line on the physical lottery ticket, which could lend to more people just going with simple geometry.

The chances of each winning number being a multiple of nine is improbable, but any other individual number selection is equally improbable.

So if you assume one improbable event, the winning lottery numbers, paired with a less improbable event, the players’ selection of their own numbers, it doesn’t seem that unbelievable, statistically speaking.

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Odds of winning the big Mega Millions prize

With tonight’s Mega Millions jackpot estimated at $1.28 billion, you might be wondering what the odds of winning are, even if you know the chances are super slim for an individual. (On the other hand, the more tickets purchased overall, the greater the chances that someone in the country wins.) For The Washington Post, Bonnie Berkowitz and Shelly Tan made a playful quiz to test your perception of 1 in 302.6 million.

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Lottery hacking, winning millions

I always love a good lottery hacking story. Jason Fagone for The Huffington Post chronicles the winnings of Gerald and Marge Selbee, a retired couple from a small town in Michigan. It is a story of probabilities, expected values, and arduously buying a lot of tickets to maximize profits.

That’s when it hit him. Right there, in the numbers on the page, he noticed a flaw—a strange and surprising pattern, like the cereal-box code, written into the fundamental machinery of the game. A loophole that would eventually make Jerry and Marge millionaires, spark an investigation by a Boston Globe Spotlight reporter, unleash a statewide political scandal and expose more than a few hypocrisies at the heart of America’s favorite form of legalized gambling.

I think it’s every statistician’s fantasy to crack open a lottery’s flaw using the numbers. No? Just me? Okay, whatever.

The most interesting part though is that the loophole didn’t seem to be that obscure. Selbee just needed a bit of knowledge about big numbers, a pencil, and a napkin to crunch on. Are there more games out there like this? Do I need to start playing the lottery?

See also the statistician who cracked a scratch lottery code and the other statistician who won the lottery four times.

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Looking for improbably frequent lottery winners

After hearing the story of reporter Lawrence Mower, who discovered fraudsters after a FOIA request in Florida, a group for the Columbia Journalism Review and PennLive looked to expand on the analysis.

Intrigued, we wanted to chart new territory: to find out whether these repeat winning patterns exist across the country. We decided to submit public records requests in every state with a lottery—an adventure in itself given that FOIA laws vary significantly by state. In all, we sent more than 100 public record requests to lotteries for information about their winners, game odds, and investigative reports. Getting those records wasn’t simple, as we outline below.

See more details.

This should be turned into a class project for Stat 101 courses.

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Nerdy Powerball FAQ

The Powerball FAQ was most likely written by a slightly annoyed statistician. You'd think the FAQ would be full of legalese and vague statements, but it reads more like notes from the know-it-all in your Stat 101 class. The answer to, “Your odds and probabilities are wrong.”:

Are not. Sure, the odds of matching 1 red ball out of 26 are 1 in 26, but we are not giving the odds for matching a red ball. We give the odds for winning a prize for matching one red ball ALONE. If you match the red ball and one or more white balls, you win some other prize, but not this prize. The odds of matching one red ball ALONE are harder than 1 in 26 because there is some risk that you will also match one or more white ball numbers - and then win a different prize.

Some persons who enjoy statistics (they do really exist) will come up with odds of 1 in 17 billion for the jackpot prize. Remember that you don't need to match the numbers in exact order - we use combinations to determine the probabilities for the first five white balls and not permutations.

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Try to win the lottery

Lottery simulator

The Powerball Lottery is big news in the United States right now. The jackpot sits at $800 million, they draw the numbers on Saturday, and it's likely someone is going to suddenly be rich soon.

This naturally comes not long after the Gaming Commission changed the rules last October, which increases the odds of winning something but decreases the odds of winning the jackpot.

In case you're not familiar with the rules of the game: Players choose five numbers and one “Powerball” number. The first five numbers used to range from 1 to 59 but since October, they range from 1 to 69. Conversely, the Powerball number used to range from 1 to 35, but now they're 1 to 26. That shifts the odds of winning the jackpot from about 1 in 175 million to 1 in 292 million.

So something very unlikely, became much more unlikely. Hence the current big jackpot.

Screw the odds though. That's a lot of money and will buy you enough tacos to make your head spin. Jon Schleuss for the Los Angeles Times provides a simulator to try your hand.

Lose your paycheck for pretend here.

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Billion Dollar Scam: Why you should play the lottery instead of going to H&R Block

H&R Block is running an aggressive campaign under the rubric “Get Your Billion Back” trying to convince taxpayers to come and have their taxes done by one of their “Tax Professionals”. Their pitch is that “this is how much money is left on the table when people do their own taxes”.

1 Billion Dollars

On their site and in TV ads they provide all sorts of information meant to wow you about how much $1 billion dollar is.

It’s $500 on every seat in every profession football stadium in America!!!

It’s 869,565,217 Bags of Chips!!!

It’s a stack of money that reaches the Van Allen Belts.

It sure sounds like a lot of money. But you know what it really is? It’s bullshit.

The $1 billion they are referring to is “left on the table” by the 56,000,000 Americans who do their own tax returns. The math is simple. That’s an average of less than $20 back per return. Since it costs an average of $198 to have H&R Block do your taxes [1], that is a completely miserable return – a pay out of roughly nine cents for every dollar spent. Or put another way, if everybody who did their own taxes went to H&R Block to have their taxes done, they would spend over $11 billion dollars. That’s 9,641,739,130 bags of chips!!

This is just an insane financial proposition. I don’t know what fraction of people get more than $198 back, but it has to be pretty small. H&R Block say that 1 in 5 people get more money back than they would have if they’d done their taxes themselves. So these people average around $100 back. Even if you’re one of these “lucky” people, you still net negative on average. Plus the upside has to be pretty small – and is capped on the high end by the amount you actually paid in taxes.

Now let’s compare that to everybody’s favorite “bad deal” – the lottery. A typical lottery in the US pays back around 60 cents for every dollar collected (this number varies depending on where you play and which game you play – some states are as low of 50 cents on the dollar, some as high as 70 [2]). But as bad a deal as this is, you are still doing six times better than if you go to H&R Block! Plus, there’s actually the prospect – albeit a small one – of a really big payoff.

If all of those 56,000,000 tax filers used their $198  – or let’s make it $150 so they can buy tax prep software to do it at home – and bought Powerball tickets throughout the year (there are around 100 draws so they’d buy 1 or 2 tickets per draw), that would be around 4.2 billion tickets. With odds of hitting the big jackpot at around 1 in 175 million, that means that 24 of these people would win in jackpot every year – at an average of around $140 million per jackpot (at this point they would actually need to see a tax professional). And another 800 people would win $1,000,000 prizes.

None of this makes playing the lottery a good investment  of course – but it’s a hell of a lot better than falling for H&R Blocks scam that is cynically trying to take advantage of most American’s innumeracy to take billions of dollars off your table.