Q: There are n tickets in the lottery, k winners and m allowing you to pick another ticket. The problem is to determine the probability of winning the lottery when you start by picking 1 (one) ticket.
A lottery has N balls in all, and you as a player can choose m numbers on each card, and the lottery authorities then choose n balls, define L(N,n,m,k) as the minimum number of cards you must purchase to ensure that at least one of your cards will have at least k numbers in common with the balls chosen in the lottery.
S: This relates to the problem of rolling a random number from 1 to 17 given a 20 sided die. You simply mark the numbers 18, 19, and 20 as "roll again".
The probability of winning is, of course, k/(n-m) as for every k cases in which you get x "draw again"'s before winning, you get n-m-k similar cases where you get x "draw again"'s before losing. The example using dice makes it obvious, however.