# 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.

1. Ted says:

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.

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2. Eric M. Jones says:

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.

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3. EP says:

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.

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4. Garvit Sah says:

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.

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5. Steven Surowiec says:

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

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6. Zach says:

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.

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7. Rich says:

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

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8. MikeM says:

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.

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