Suppose that a lottery ticket has probability p of being a...

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 $3 p-3 p^{2}+p^{3}$, 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.

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 $3 p-3 p^{2}+p^{3}$, 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.

Key Concepts

Binomial Distribution

The binomial distribution describes the number of successes in a series of independent trials, each with the same probability of success. It is characterized by two parameters: the number of trials and the probability of success in each trial. This concept allows us to calculate the probability of obtaining any specific count of successes by using the formula involving combinations and the success/failure probabilities.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a combinatorial method used to compute the probability of the union of several events, especially when these events are not mutually exclusive. This principle corrects for the over-counting that occurs when simply summing individual probabilities by subtracting the probabilities of all pairwise intersections and adding back in the probability of the triple intersections, and so on.

Complement Rule

The complement rule in probability is a fundamental concept stating that the probability of an event occurring is equal to one minus the probability of the event not occurring. It is particularly useful when the direct computation of an event is difficult, but its complement is easier to calculate, as it simplifies many problems in probability.

Small Probability Approximation

When dealing with small probabilities, higher-order terms in the calculations (such as squared or cubed terms) become negligible compared to the linear term. This approximation allows for simpler expressions and insights into how the probability scales, providing a quick estimation that is often practically sufficient, especially in cases where the precise values of small probabilities do not impact the overall outcome significantly.

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