What are the odds of winning this bingo game? I'll explain the game real quick, it's called Pick 8.
You get a sheet of paper, containing 3 rows of 8 boxes. You fill out each row with numbers 1-75, in any order, with no duplicates. The bingo caller then starts calling bingo numbers.
If you get a row filled during the first 20 numbers called, you win the jackpot, which is usually around \$7000. The first person to win after 20 balls are called wins \$500.
I'm interested in the jackpot. Mainly, how many sheets would I need to buy and fill out with an 8-number combination to ensure ~70% probability of winning in the first 20 numbers called? Each sheets costs \$3, so it's \$1 a row.
I started by calculating how many 8-number combinations there are in a pool of 75 (where order doesn't matter). Apparently that's called a binomial coefficient, and it comes out to ~16.8 billion combinations.
But I don't where to take it from there - how do I take into account the fact that I get 20 chances to get an 8-number combination correct? If I can figure out that, then I can multiply those odds by 3, and then multiply it even further to cover 70% of the possible combos, to see how many sheets I'd need to buy.
 A: The probability of winning depends on the way you choose the rows. To have a vague idea of the numbers you can look at what happens if each row is filled at random (this is clearly not optimal, for instance it includes the possibility that you use two identical rows).
For a single row, the probability of winning is the ratio of draws of 20 numbers out of 75 that include the 8 selected numbers in the row to the total number of draws of 20 numbers out of 75, i.e.
$$
p=\frac{75-8 \choose 20-8}{75 \choose 20}= \frac{1}{133\,929}.
$$
If you play $n$ rows, each independently and randomly selected, then the probability of winning is
$$
1-(1-p)^n.
$$
If you want that to be $70~\%$ you solve for $n$:
$$
1-(1-p)^n=.7 \rightarrow n=161\,246.
$$
This is clearly an upper bound, as a better strategy should do better that selecting rows at random.
Edit: details on the computation of $p$. Once you have chosen your 8 numbers, the number of draws of 20 numbers out of 75 that will make you a winner is the number of ways to pick your 8 numbers exactly (just one way) and then to pick the remaining 12 numbers out of the remaining 67 numbers: $1\times{67\choose12}$. The total number of ways to pick 20 numbers remains ${75\choose20}$. The probability is the ratio of favourable draws to total draws, thus the above formula.
