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# A +EV Lottery?

Lotteries are typically very stupid. That is not because winning is unlikely, but because they typically have drastically negative expected values. However, over the next few minutes there is an American lottery, Mega Millions, that some claim has an anomalously positive expected value. According to the game's rules, Mega Millions tickets cost \$1.00 per play. Players may pick six numbers from two separate pools of numbers - five different numbers from 1 to 75 and one number from 1 to 15 - or select "Easy Pick". You win the jackpot by matching all six winning numbers in a drawing. The jackpots start at \$15 million and grow by a minimum of \$5 million per draw each time the jackpot rolls.

The drawing is scheduled for this evening at 11 PM eastern standard time. The jackpot is estimated at \$636 million. While winning is extremely unlikely, it appears to drive a positive expected value for each \$1 of cost:

Why is this opportunity available? Is the demand for tickets dynamic enough to account for changes in the jackpot? Are huge jackpots such as tonight's occasionally undersubscribed? If you happen to read this, play, and win, I would recommend taking the annuity instead of the upfront cash option, based upon the following chart; the horizontal axis shows the number tickets sold, in millions, and the vertical axis shows the expected value in dollars of a ticket, based on the likelihood of a split, and the size of a split jackpot.

Forget about luck. Instead, I would prefer to wish you a good expectancy!

Is there a catch? Sadly, there are catches. The first is taxation. Federal (often combined with state) taxes reduce the jackpot value by over a third.

In addition, lottery sales data indicates that large jackpots, which drive heightened press interest, are fully subscribed.

Based on the hysteria around massive jackpots, increased demand actually lowers EV/ticket:

Finally, according to computer scientist and mathematician Bill Butler, there is a significant impact from split pots:

Since tonight's jackpot will probably not go to a single winner and any eventual winners will have to pay a large tax bill, the EV is reduced from \$1.63 to about \$0.64 after taxes. Also, the payoff structure is hugely propped up by tiny changes of huge windfalls, so that it is extremely likely that you will not win. If you play, the EV indicates that you are playing a losing game and the payoff structure indicates that you will not be able to play for long since your bankroll will be quickly consumed.

What do you do if you are comped a ticket? I recently attended a wedding at Lake Tahoe and we stayed at a hotel that had a casino. Various introductory bets came with the room. It might be nice to know how to make a less bad bet, even when it is not +EV. The key is to bet underused and unlucky numbers so as to reduce the chance of splitting a pot. Overused numbers include 1, 7, 13, 23, 32, 42, and 48. Underused numbers include 26, 34, 44, 45, and 46. Avoid number combinations that can form dates (pick two digits above 12, then two above 31, then two outside of common year ranges of 1900-2013). Despite this number system showing some merit, it still appears to be -EV in the US.

I am interested in doing more work on Chinese lucky numbers. Culturally, many Chinese are heavy gamblers and are also fascinated by lucky numbers (0, 6, 8, and 9 appear to be overused, while 4, 5, and 6 appear to be underused). Other number combinations sound like insults or vulgarities that may also be underused. This may be profitably exploitable in Chinese games.

For further reading on lotteries and their expectancies, you might like Finding Good Bets in the Lottery, and Why You Shouldn t Take Them.