[bumbledraven]

Mark Kon, a professor of math and statistics at Boston University, calculated that a bettor buying even $10,000 worth of tickets would run a significant risk of losing more than they won during the July rolldown week.

A $2 Cash WinFall ticket allows you to pick a set of 6 distinct integers from the set {1..46}. The lottery commission then picks six distinct integers at random from the set {1..46}, effectively by drawing numbered balls without replacement from an urn. The payouts for May 9, 2011 were $24821 for a match-5 (a ticket that has 5 numbers in common with the set chosen by the lottery), $824 for a match-4, and $26 for a match-3 (http://goo.gl/gSTsF). It's easy to show that buying a ticket on that day had a positive expected value.

[Edit to show expected value calculation: According to http://www.masslottery.com/games/lottery/cash-winfall.html, the probability of a match-5 is 1/39028.41, the probability of match-4 is 1/800.58, and the probability of a match-3 is 1/47.40.

Then the expected value of a $2 ticket on May 9, 2011 was $24821/39028.41 + $824/800.58 + $26/47.40 = $2.21.]

$10,000 will buy 2,000 [Edit: oops, actually 5,000. Also fixed in what follows] Cash WinFall tickets. When you can only buy a few tickets in a lottery like this, the distribution of the numbers on your tickets greatly affects your chances of coming out ahead. For instance, if you buy 5,000 copies of the ticket {1,2,3,4,5,6}, your chances of coming out ahead are much worse than if you buy 5,000 tickets with a bunch of different numbers, although of course your expected winnings are the same in each case, provided no one hits the jackpot.

It's actually an open problem to create a set of 5,000 Cash WinFall tickets that maximize your chances of coming out ahead. Indeed, a Boston-area employer (that's still hiring!) posted a hiring problem based on this for several years.

[mhb]

The rules also mention:

If the total prizes won for any drawing, exceed 200% of the net sales for that drawing the prize amounts will be based upon a formula detailed in the Rules and Regulations or Administrative Bulletins issued thereunder.

which has to add a little FUD into the thinking.

[bumbledraven]

Yes, although historically it's clear that the payouts make it profitable to buy tickets on rolldown days, the actual formula used to calculate the payouts was opaque. I heard that one of the lottery syndicates actually made a request under the Massachusetts Public Records Law to get the lottery commission to reveal the formula, which they eventually did.

[mhb]

$10,000 buys 5000 tickets, but I'd be interested in seeing how the positive expected value is easily shown.

[mhb]

Right. I was actually interested in seeing how the probabilities were calculated which doesn't seem as easy.

[bumbledraven]

http://en.wikipedia.org/wiki/Lottery_mathematics has a worked out example for a lottery that involves picking 6 distinct numbers from {1..49}. It would be straightforward to adapt that to Cash WinFall.

[nonymousse]

Which company has this as a hiring problem?

[bumbledraven]

Past tense. They don't have that hiring problem posted any more, but it was for mathematicians/software developers in the R&D group. If you're asking because you have an approximate solution, let me know (email address in profile) and I can put you in touch with people there who would be very interested to see what you've come up with.

[randallsquared]

Probably he means ITA Software, which was bought by Google last year.