# A Lottery for People Who Are Good at Math

As investment schemes, state lotteries are about as sound as a **Bernard Madoff** venture.

But at least one lottery might be worth it — if you do the math.

When the jackpot for the match-six-numbers Massachusetts Cash WinFall tops $2 million without a winner, the prize money rolls down to the lower-tier winners, dramatically increasing the value of a win for people who match 2, 3, 4, or 5 numbers.

**Mark Muir** did the math, and figures that if you buy enough tickets under the roll-down scenario, you’re statistically guaranteed a return on your “investment.” The numbers start to get substantial the more tickets you buy. For example, he figures that when you buy $10,000 worth of tickets under roll-down conditions:

[Y]ou could expect 732 tickets to match 2, 105 to match 3, 6 to match 4, and a 12.8 percent chance to match 5. The expected value in this scenario is $14,280.

If you are observant, you will note that for this payout structure, the ratio of investment to winnings is a constant 42.8 percent return. That is, for every dollar invested, you can expect $1.428 in return over the long term.

Alas, the scheme does come with some risks, which Muir details at his Million Dollar Idea Guy blog.

## Ted

There is one major problem with this idea. The number of people playing on any given week affects the odds, and there is no way to know how many people are playing. That means that on any give week, you can never be sure that the bet is +EV.

## Eric M. Jones

So Dube...

You think the Massachusetts lottery scam artists didn't changed the odds ten minutes after this was first disclosed?

I've got a lovely 19th century gothic bridge from Brooklyn to Manhattan I can sell you.

## EP

Math Horizons, the official publication of the Mathematical Association of America, had a story in their February 2007 issue about detailing a similar lottery senario in New York. In "Mixing a Night out with Probability… & Making a Fortune", Kari Lock describes how, in a promotion, the state lottery doubled the payoff for a keno-style lottery game. What they didn't realize was that this raised the expected return value above 1. Since this lottery game didn't split winnings and there wasn't the doomsday senarios presented in the link above, it was easy enough to take advantage of. Fortunately for the lottery commission, only one set of nerds followed through. They spent the whole night at a bar buying tickets from a dispenser. They never bothered to check their numbers, they just kept buying tickets all night long. After they tallied it up and chashed in, they retured and bought new houses.

## Garvit Sah

One of the biggest problem with lotteries or any gambling is that expectated value is not the payoff. I think the reason casinos make money is because of the facat that few can go on playing for long enough to reach that expected value. Whether one can make money out of such a strategy also depends, to some extent, on the individual's bank roll - those who start with a higher bank roll should have more chance of nearing the expected value.

As Mark points out, if more and more people start adopting this strategy, the payouts will reduce. This makes this strategy similar to an arbitage situation where few can make money and that too for a short span of time.

## Steven Surowiec

Until you take into consideration the taxes the government will charge you for your winning.

## Zach

This reminds me a scheme I heard about over 10 years ago. In that case, the Powerball jackpot got so big that you could buy every single combination of numbers and still turn a profit if you were the only winner. A group of lawyers consulted a statistician to double-check their theory. They were correct in their initial assumption, but the statistician pointed out two potential downfalls in the strategy. Number one was the obvious...if there is more than one winner, the jackpot is split. More people play when there is a higher jackpot. The odds of buying a single ticket and winning are incredibly small. If you buy out all combinations, though, you are betting on nobody winning. With more and more participants, the chances of somebody winning grow. You are betting against somebody winning, and putting down the hundreds of millions of dollars on that bet is not a good risk-reward prospect. Especially when your own actions will be nearly doubling the pot and making participation even more attractive for the guy on the street.

The second statistic the guy generated was an examination of the logistics. A store can only print tickets so fast. To get one of each, he estimated how many people would have to go to a different store and spend every hour the store was open getting one ticket after another. With the number of people involved, he estimated a large percentage of tickets you were supposed to buy in order to corner the market would go unbought. There is just too much room for mistakes, and if the winning combination is a ticket you mistakenly didn't buy, you lose the entire "investment"

So it's easier to see why the Powerball buy-out scenario is treacherous. Take the lessons learned in that scenario and reconsider how a person might go about buying 10,000 tickets. Then keep in mind that if you could successfully pull it off, as Muir points out, other people will get in on the deal and increase the chance that somebody ruins everything by winning the jackpot.

To relate this to an earlier post, though, consider that what you are buying is not an investment, but entertainment as you dream of what you would do with unlikely winnings. I live in Massachusetts, and you can bet that I will buy a single ticket next time the jackpot is about to roll. Having slightly better chance at winning will make my entertainment factor a little more enjoyable. That's worth $2.

## Rich

Another risk is that the State could label you as a terrorist trying to launder money and refuse to pay out.

## MikeM

He didn't FINISH the calculations! If the chances of someone hitting the jackpot effect the EV for that play, that needs to be included in the calculation. Similar if the payouts slide downward if there are many winners.

Either these plays have a positive EV or not. You can't stop the math half-way and claim a positive EV but for additional factors.

## Chris

I went to grad school at a certain highly distinguished technology institute in Massachusetts. There is a group of former students who have been doing this exact thing for years - they have a lottery pool set up, and track the projected payout of the cash winfall lottery to determine when the $2million is expected to be rich. As I understand it, they spend all day buying tickets at several locations throughout the state, and enjoy about a 10% return after taxes.

## Mike

The return is 1.42... IF no one wins the jackpot that day. IF it actually crosses the $2,000,000 threshold.

And IF you count the $2 free bet as $2 cash. Which is ludicrous, given that in a non-rolldown scenario, your return is -65%.

And, as someone above mentions... you lose a healthy chunk to taxes.

## Kevin

I looked at this with the Canadian 6-49 lottery (nationwide, the jackpot rolls over from one draw to the next). For the first couple of roll overs, the increased jackpot size increased the expected value of winning. But after around three roll overs, the odds of splitting the prize (since large jackpots draw more attention) increased at around the same rate as the size of the prize, effectively arbitraging out the marginal gain. I was surprised at how efficient the market seemed to be, actually.

This Mass. lottery seems to split the prize pool in these "Winfall" situations, since they have unusual return amounts, so my guess is the same problem applies here.

The other elements of possible strategy, however, are in avoiding popular numbers (e.g. 1-2-3-4-5-6; anything under 31 since many players play with dates), which partially counteracts the split effect, and in pooling money with more players -- in effect, taking advantage of the diminishing margin of return.

## JS

This strategy is similar in at least one respect to a martingale strategy - being profitable on an expected value basis. The trick is you must be able to handle the swings.

## William

I notice that early on, the payoffs for matching 3/6 were $66, $68, $86, $77, $62, for an average of $71.80 for the first five times the lotto had a positive payoff.

The most recent 5 (not including the time one man won the jackpot), the payoffs for matching 3/6 were $29, $27, $37, $30, $28, for an average of $30.20.

I'm thinking that recently, a lot of people have been gaming the system in exactly this way, and they are driving down the profits. This also makes it less likely that the $2,000,000 threshold won't be met (it hasn't happened since mid-2005), but makes it more likely that someone will win on a $2,000,000 jackpot day (it first happened in mid-2007).

Even so, the worst expectation value of profits yet for a favorable betting day was +$0.32 per dollar bet (assuming you nobody wins the pot). Not bad for a system that can be gamed about 7 times per year.

## William

Also, there have been 30 of these "big day" lotto days, and only 2 of them have gone awry. So, with a 1/15 odds of getting -33% returns, and 14/15 odds of getting + 32% returns, you're looking at like 28% returns. Still not bad!

Even assuming the government shaves off 1/3 of the winnings, you're looking at like 18% returns. You just need at least $11,300 or so to bet, and a total of $15,600 or so (the excess is so that you can kick in some money if the return goes negative). You'd still be able to double your money each year. Minus expenses, of course.

## William

The next question you want to ask is, how much time does it take (on average) to buy 5,000 tickets? (include travel time). How much time does it take (on average) to read all of those 5,000 tickets? And lastly, how much is your time worth?

I'm thinking a reasonable estimate is, if you live above a place that sells these tickets, 2 tickets / minute to buy, throughout the day (including breaks). So, in a 12-hour day, you might be able to buy 1,440 tickets. You'd need 3 buddies to buy 6,000 of them, all of you together.

Also, assume that you can check 6 tickets per minute. So, you're looking at spending 16 hours on this project, and buying 1,440 tickets (call it 1500, for $3,000), with maybe an 18% return after taxes. So you earn $540 in exchange for 16 hours of work, or about $33.00 per hour.

That's about the hourly rate of someone who makes $65,000 per year, assuming they get no benefits.

## DaveV

In 1992 an Australian betting syndicate tried to buy every one of the 7 million number combinations to try to win the $27 million jackpot in Virgina. The only bought 5 million tickets, but won the jackpot anyway, and they were the only winner. After that, Virginia changed the rules so that players now have to fill in a card for each and every combination that they wished to buy.

http://query.nytimes.com/gst/fullpage.html?res=9E0CE0D61439F936A15751C0A964958260&sec=&spon=&partner=permalink&exprod=permalink

## Lollie Dot Com

Speaking of million dollar ideas.... I've got an idea for a freakonomics romance reality show.... Can't Match This!! It's a show where contestants vie to see who is the hardest to find a mate for based on their statistical deal breakers. For example, I'm overweight, that knocks out what? 85-90%, I don't mind a doobie now and then, what percentage does that drop my numbers to? I have a high i.q. but a seventh grade education. What percentage of intelligent men will date someone with a seventh grade education? I'm a professional psychic.... and omg, it gets better, I'm not into new age crap. The crystal holding, spell casting, talking to dead people, spirit guides, none of it. New agers, the one area where I could really gain some percentage.... ugh. I wouldn't believe in psychic ability at all if I didn't have any of it.

That would be a great show because a lot of people think they're harder to match than someone else, but statistics would show who was right. The show would help people realize which things they could most comfortably improve or eliminate with the highest returns to their love life. It would make money for you plus entertain, educate and benefit one and all.

And if I was a contestant, of course, I'd win. So I'd really, really love for you to do that show. :) The show could end with the reassuring reminder, it only takes one.

Although I do believe you'd find I'm more likely to be incinerated in the afterburn of the Infinite Improbability Drive than find mine. http://www.earthstar.co.uk/drive.htm

And where will I be after I finally walk the remainder of my excess weight off? Will I have a shot then?

You can answer these questions that many people ask, with a show like "Can't Match This!!"

## David

As many of the posters above have mentioned, this is not a guaranteed payoff (for both probabilistic and logistical reasons).

What I don't get is what "statistically guaranteed" means. As a term it pops up every now and again and really has absolutely no meaning. If something is guaranteed, it is guaranteed independentg of statistics. If it is highly likely, then it is highly likely with a certain probability of occurring.

Lazy language like this is a big part of the reason statistics are treated with disdain. Expected values are dangerous animals on their own. Put them in a cage with variance, skewness, kurtosis and higher moments and then let's see who eats whom.

## Joe C from Austin

I can't find the article I once read about a Jai Alai championship, but sometime back (80's or 90's) a group of gamblers raked in well over a million dollars when they recognized that an investment of about $250K would cover all remaining outcomes in a tournament.

There was an outcry in the gambling community that without risk this wasn't gambling, but I suspect that there was an equivalent rejoicing in the math community that it WAS math. Stay in school kids.

## Bryan Larsen

Lotto 6/49 in Canada reaches an expected payoff close two or even above $2 on a $2 ticket when the jackpot goes above 30MM. Knowing the exact payout requires knowing how many people bought tickets, so you can accurately determine the odds of splitting the top prize.

I did the calculation on a past 54MM draw, and the payout turned out to be $2.94. If a group purchased every ticket, the payout would have dropped to $2.50 because they would have increased the number of people buying tickets. If 2 groups did the same thing, then the payout would have dropped below even.

Regarding the above comment on 6/49: yes, they do change the odds above 30MM, but the money gets moved down into the match 5 and match 4 categories, so the expected payout is not substantially different. They keep a little more, but a large portion of the money you win is from previous draws that didn't win rather than the new money pumped in for the current draw.