Calculate Probablilty of winning Lottery/Bingo over Successive Draws

I am normally pretty good at figuring out probabilities but this has me stumpped.

I am playing in a fundraiser where people select "lines" of 6 numbers (between 1 and 49) just like a lotto 6/49 ticket. Then the public draw occurs and you look at your numbers and if any match with those drawn, you cross-off your matching numbers. Then the next draw happens a week later and you do the same thing. So you are accumulating crossed-off numbers sort of like bingo. It may take several draws, but the first person or people to have all six of their numbers crossed-off wins (and they get a prize).

I want to be able to calculate and graph the increasing probability points for each sucessive draw. I know for one line, after the first draw it is "49 choose 6" or 49!/43!/6! or 1 in 13,983,816. I am have trouble formulating the next probabilities, I think because you get to replace the balls after the first 6. The simple and/or scenarios (multiply and add) don't seem to work.

The second complicating factor in this question, is how the number of players (or lines) in the group effects the probability of someone winning. This may be a tougher question but it may be just the "OR" scenario so you just add up or multiply by the number of lines attempting to win. Please confirm? It is safe to assume that no line will have the same 6 numbers.

I have modelled this in Excel and can tell that with a group of 12 people, someone tends to win 50% of the time over a thousand± rapid iterations after only three draws and after 5 draws there is a winner almost every time over about 500 iterations. It is not possible to translate this into rough probabilities, however. It gives more of a gut feeling. Now I want to see the proof!

Thanks for any help. Cool question don't you think? I love math.

Cheers!(Whew)

Thanks for any guidance towards an answer.