(s) Players A and B take turns in answering trivia...

(s) Players A and B take turns in answering trivia questions, starting with player A answering the first question. Each time A answers a question, she has probability $p_{1}$ of getting it right. Each time B plays, he has probability $p_{2}$ of getting it right. (a) If $\mathrm{A}$ answers $m$ questions, what is the PMF of the number of questions she gets right? (b) If A answers $m$ times and B answers $n$ times, what is the PMF of the total number of questions they get right (you can leave your answer as a sum)? Describe exactly when/whether this is a Binomial distribution. (c) Suppose that the first player to answer correctly wins the game (with no predetermined maximum number of questions that can be asked). Find the probability that A wins the game.

(s) Players A and B take turns in answering trivia questions, starting with player A answering the first question. Each time A answers a question, she has probability $p_{1}$ of getting it right. Each time B plays, he has probability $p_{2}$ of getting it right.
(a) If $\mathrm{A}$ answers $m$ questions, what is the PMF of the number of questions she gets right?
(b) If A answers $m$ times and B answers $n$ times, what is the PMF of the total number of questions they get right (you can leave your answer as a sum)? Describe exactly when/whether this is a Binomial distribution.
(c) Suppose that the first player to answer correctly wins the game (with no predetermined maximum number of questions that can be asked). Find the probability that A wins the game.

Key Concepts

Binomial Distribution

This is a probability distribution that describes the number of successes in a fixed number of independent trials where each trial has the same probability of success. It provides the formula P(X = k) = C(n, k) p^k (1 - p)^(n – k) for k successes in n trials, which is directly applicable when modeling the number of correct answers by a single player over a set number of questions.

Convolution of Independent Random Variables

When dealing with the sum of successes from two independent binomial processes with potentially different parameters, the overall distribution is obtained by the convolution of the two separate binomial distributions. This method involves summing over the joint probabilities of all combinations that yield a particular total number of successes, and the resulting distribution will only simplify to a binomial form if the success probabilities for both processes are identical.

Geometric Series in First Success Problems

In scenarios where the event of interest is the first occurrence of success (such as winning on the first correct answer), the problem can be modeled using a geometric series. Each pair of turns (one for each player) can be treated as a cycle where both might fail, leading to a repeated independent structure. The probability of the first success is then computed by summing the probabilities over these cycles, resulting in an expression that is derived from the sum of an infinite geometric series.

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