It’s a commonly held misconception that if you play the lottery n times, and the probability of winning each time is 1/N, then your chance of winning at least once is n/N. That’s true if you buy n tickets in one week, but not if you buy a single ticket in each of n independent weeks. Let’s explore further.
a. Suppose you play a game n independent times, with P(win) = 1/N each time. Find an expression for the probability you win at least once.

b. How does your answer to (a) compare to n/N for the easy task of rolling a 4 on a fair die (so N = 6) in n = 3 tries? In n = 6 tries? In n = 10 tries?
c. Now consider a weekly lottery where you must guess the 6 winning numbers from 1 to 49, so N = (49/6). If you play this lottery every week for a year (n = 52), how does your answer to (a) compare to n/N?

d. Show that when n is much smaller than N, the fraction n/N is not a bad approximation to (a).