Suppose I am a high-volume broker aiming to make some money on a state lottery. In this lottery, six balls are drawn from a population of (let's say) 50, without replacement. A ticket is a choice of a size-6 subset of 1,2,..50.
The prize structure of this lottery is such that the jackpot alone doesn't impart much value to the ticket. But it turns out that lesser prizes are sufficiently large relative to their probability that the ticket has a positive expected value, which is why I'm buying a lot of tickets in the first place. For instance, I can expect to get a pretty substantial return from tickets which match 4 of the 6 numbers drawn by the lottery.
There is a substantial literature related to the "Turan problem," which asks: what is the minimum number of tickets I need to purchase in order to guarantee that one of my tickets matches 4 of the 6 numbers in the lottery?
My question is somewhat different. Let's say I have enough capital to buy a fixed number N of tickets, large in absolute terms but small relative to 50 choose 6. Then my expected gain is fixed. But of course as a wise investor I may want to minimize the variance of my winnings.
Thus my question.
If the random variable X is the number of (4 out of 6) wins among my N tickets, how small can I make Var(X) by judicious choice of ticket purchases?
(Of course, the same question applies for (k out of 6) where k=2,3,5.)
By the way, in case the setup seems unrealistic, let me add that the reason I'm asking this is that the situation described here actually happened, and I'm trying to reverse-engineer what the broker's risk-minimization strategy must have been, and assess whether it was worth it.