Why a Mega Millions Ticket is a Good Bet, But… (Revised 03.30.2012)

The Mega Millions multi-state lottery is now advertising a jackpot of $640 million for tonight’s drawing. The cash value of the jackpot is $460 million, a more accurate estimate of the present value of the prize. The jackpot has accumulated because there has been no winner for 18 consecutive drawings – since January 24th. This means that those of us who buy Mega Millions tickets for the first time this week have been subsidized by the millions of players who contributed to the jackpot over the past two months.

The probability of correctly selecting all six numbers for Friday’s drawing is one in 175 million. It is likely that there will be multiple jackpot winners. Even after taking this into account, the expected value of a one dollar Mega Millions ticket is about 90 cents.

About 18 cents of every dollar collected from Mega Millions sales are used to pay consolation prizes. Another 32 cents of each dollar goes into the cash jackpot. The cash value of Friday’s drawing is estimated to be about $200 million higher than in Tuesday’s drawing. In other words lottery officials expect about 625 million lottery tickets to be sold between now and Friday.

If lottery officials are correct there is a 96% chance that someone will pick the winning numbers on Friday, and the expected number of winners, conditional on the jackpot being awarded, is about 3. This means that even after splitting the jackpot, the expected value of a Mega Millions ticket is about 90 cents (including 18 cents in expected consolation prizes).

In general state sponsored lottery games have low expected values for players, relative to other forms of gaming. Lotteries pay out about half of their proceeds in prizes, on average. The rollover nature of the jackpot in state lotteries means that a lottery ticket will have a high expected value on occasion. Until the recent surge in sales, this week would have been the highest expected value for a Mega Millions ticket in the history of the game.

Finally, because the jackpot will be taxed at a fairly high rate a Mega Millions ticket is not a good investment, as Ben Casselman of the Wall Street Journal has noted. After taxes the expected value of a Mega Millions ticket is now much less than $1, unless the top marginal tax rate falls substantially over the next twenty years, which is doubtful. However most gambles with long odds, such as a super trifecta bet at the racetrack, have a pre-tax expected value that is lower than the cost of the wager. So the Mega Millions is an unusually good bet for people who enjoy gambling small amounts at long odds, but it would be a bad investment.

Comments

  1. what about taxes?

  2. Can we see your math, like this site shows: http://www.durangobill.com/MegaMillionsOdds.html

  3. If lottery works the same way as, say, poker winnings, then that means you can deduct the cost of the ticket from your winnings, up to the total winnings amount. Thus, the post-tax odds becomes somewhat irrelevant.

Trackbacks

  1. […] problem, though, is that even if you win, there’s no guarantee you’d be the only winner. According to economist Stephen Bronars , lottery officials figure there’s about a 76% chance of someone picking a winner in tomorrow’s […]

  2. […] It’s likely that there will be multiple winners (2.5 according to the math), but even so, the … Still, the probability of zero winners is about 10%. [url] […]

  3. […] It's likely that there will be multiple winners (2.5 according to the math), but even so, the expect… Still, the probability of zero winners is about 10%. [url] […]

  4. […] the jackpot gets split up. Economist Stephen Bronars estimates that there are likely to be two or three winners. So let’s say there are two winners. They each get $195 million. Still a healthy profit on […]

  5. […] there are likely to be two or three winners. so let’s say there are two winners. […]

  6. […] But in strict net-present-value terms, the lottery still wasn’t a good investment. (Though as economist Stephen Bronars noted at the time, the expected return was better than in many other long-odds […]

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