A Chicago area man won the lottery for the *second* time. The Chicago Tribune reports:

Scott Anetsberger duplicated his $1 million win of nine years ago in the same instant Merry Millionaire game, lottery spokesman Mike Lang said.

Despite long odds, Anetsberger isn’t the first two-time $1 million instant winner. Kimberly Pleticha of Villa Park won $1 million twice in the instant Cash Jackpot game–the first time in August 2010 and the second only six months later in February.

Lottery officials could not instantly compute the odds against multiple winners, but did note there have been a dozen or more two-time Little Lotto winners over the years.

What would the odds of winning the lottery twice would be? Well, it depends on how frequently one plays the lottery.

Winning the Illinois Lottery requires picking six correct numbers, where the numbers range from 1 to 52. The odds of getting all six numbers correct is 1 in 20,358,520. It costs $0.50 to play the lottery, and there are three lotteries per week. Assuming that each lottery is independent (a reasonable assumption), one would have to play the lottery 20,358,520 times, over average, to win (using the geometric distribution). If one plays the lottery three times per week, then it would take 130,500 years to win the lottery *once* at a cost of more than $10M.

Winning the lottery *twice* can be modeled as a negative binomial random variable. Assuming that our lottery winner plays the lottery three times per week before and after winning the lottery, then it takes ~261,000 years, on average, to win twice.

Since it is only newsworthy to report additional wins by those who have already won the lottery, then we are really only interested in the odds that a lottery winner would win the lottery *again*. This is a different question. Assuming that our lottery winner continues to play the lottery three times per week, then the odds of winning again are same as the odds of someone else winning the lottery for the first time: 1 in 20,358,520 per lottery. That is, it would take our lottery winner an *additional* 130,500 years to win the lottery.

If someone plays the lottery more than three times per week, then the odds of winning go up.

Of course, many people play the lottery, so the odds that *someone* wins the lottery twice over their lifetime is much, much higher. I tell my students every semester, “Someone will win the lottery. Just not you.” If 130,500 people buy one lottery ticket per game, then there would be a two-time winner every 2 years, on average.

**Little Lotto** involves picking five correct numbers, where the numbers range from 1 to 39. It is easier to win, but it has a lower payout. The odds of winning are 1 in 575,757, which means that one is 35 times as likely to win the Little Lotto than the regular lottery. It would take 3691 years to win Little Lotto once (by playing three times per week) and 7382 years to win it twice.

Given that there have been 12 two-time winners in Little Lotto in its 23 years of existence, there there is approximately one two-time winner every two years. Given my assumptions, this would suggest that ~3691 people buy a Little Lotto ticket every time. That seems a bit low to me. But I have a head cold and maybe it has temporarily impaired my mathematical abilities.

A seven-time lottery winner’s advice for winning the lottery is to invest *more* (not *less*!) of one’s money into buying lottery tickets, as long as one can afford it. He also recommends treating the lottery as a job: the lottery is a skill, and one can improve at it after investing a lot of time. While skill plays a role in playing the lottery (identifying which numbers to pick and identifying which games have the best payoff), I’m pretty sure that this is bad advice. The expected payoff for the lottery is negative, meaning that on average, you are guaranteed to come out behind. The variance in earnings is large, meaning that over many attempts, it is possible that you can come out ahead. But given that one comes out ahead, it would be foolish to attribute one’s success to skill. But maybe I’m missing something.

For the record, I do not recommend gambling or routinely playing the lottery.

For more, read Mike Trick’s post on conditional probabilities and March Madness odds.

Related post:

January 12th, 2012 at 11:30 am

These are good reads on the mathematical limitations of some games

http://articles.boston.com/2011-07-31/news/29836200_1_lottery-tickets-claim-prizes-massachusetts-state-lottery

http://www.wired.com/magazine/2011/01/ff_lottery/all/1

I’ll email you an article from Harper’s about a Stanford PhD who shows up in Texas to buy tickets every once in a while (and usually wins)

January 12th, 2012 at 12:15 pm

Great post!

“The expected payoff for the lottery is negative, …”

As noted by Eric — in rare situations this assumption is not true, and a few math savvy (and probably somewhat risk-seeking given the “unknown” issues that could arise) people will make out quite well with a huge buy-up. From what I’ve (rarely) seen, this only happens in specific places with strange rules — like Massachusetts (as above). Fun to read about… but not my idea of a great way to generate income nonetheless…

Another issue with scratch offs is randomness — just as finding a good random number generator can be problematic in practice (what is random?), manufacturing randomness for scratch offs will always have issues too.

Ethical issues arise as well when a state has paid out all the best prizes — should they be able to sell the remaining tickets given they know the new probabilities and expected payouts are much worse? How should these prizes be reported? And so on…

January 12th, 2012 at 4:17 pm

@Eric, these are great references. Thank you!

@Paul: I knew that the expected payoff became positive on rare occasions. I must admit that I was surprised that this happened so frequently. The ethical issues that you raise are quite interesting. With today’s social networks, it is easy to determine if the best prizes have already been claimed. I wonder if that has ever affected lottery ticket sales?

January 13th, 2012 at 5:21 pm

Great post. A few thoughts to make it more clear:

1. If the event is truly random then one could still play the lottery for 130,500 years and still never win, since each lottery would be independent.

2. In order for the statement that to be true that if 130,500 people play the lottery you can expect to have a 2 time winner every two years, it would need to be the same 130,500 people.

3. Of course the probabilities can approach 100% but never actually get to 100%

January 13th, 2012 at 5:23 pm

Thanks @Cory. I looked at averages here, and you are correct: there are no guarantees.