Question: Suppose that a lottery ticket has probability p of being a winning ticket, independently of other tickets. A gambler buys 3 tickets, hoping this will triple the chance of having at least one winning ticket.
(a) What is the distribution of how many of the 3 tickets are winning tickets?
(b) Show that the probability that at least 1 of the 3 tickets is winning is 3p - 3p2 + p3, in two different ways: by using inclusion-exclusion, and by taking the complement of the desired event and then using the PMF of a certain named distribution.
(c) Show that the gambler's chances of having at least one winning ticket do not quite triple (compared with buying only one ticket), but that they do approximately triple if p is small.