Gwoof, this thread will degenerate into a lot of name-calling and other ad hominem comments very quickly. I just want to forewarn you.
I am not familiar with any closed-end formulas for this. Jimmy is mostly correct: a Monte Carlo simulation might be the best way to go. But I doubt that it requires "millions of random draws". That depends on the degree of confidence that you want for the answer. And that can be bootstrapped. I'll have more details in a little while.
(It is a possible that an analysis of a history of several hundred drawings might be sufficient. After all, that represents many hundred samplings. But I agree with Jimmy: I would prefer the structure of simulation, if only to validate conclusions from the historical drawings.)
But first, you need to buy into the concept that lottery drawings are completely random and completely independent.
That is indeed a fact. But there are many vocal participants in these forums who do not believe that. And there are others that accept the randomness of the process, but somehow they believe we can predict future draws from past draws -- or from silly arithmetic progression (which is really just a random number generator, usually a poor one).
(Note: That is not to dismiss those who like to use the probabilities of patterns of drawings in order to reduce the numbers that they select. Any such reduction is a personal choice. And all such choices are valid, although some might be better than others statistically speaking.)
If you do buy into the concept of random and independent drawings, you might as well ignore anything that Jimmy or I have to say on the matter.
PS: You also have to buy into the assertion that the random number generator used in the simulation is sufficiently well implemented. Honestly, that is always debatable. But IMHO, we have to take some things on faith until proven otherwise.