Suppose that two teams play a series of games that ends when...

Suppose that two teams play a series of games that ends when one of them has won $i$ games. Suppose that each game played is, independently, won by team $A$ with probability $p .$ Find the expected number of games that are played when (a) $i=2$ and (b) $i=3 .$ Also, show in both cases that this number is maximized when $p=\frac{1}{2}$.

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Key Concepts

Optimization and Symmetry

This concept is about finding the parameter (in this case, the win probability p) that maximizes or minimizes a certain quantity. The analysis shows that when the game is fair (p = 1/2), the series is longest on average. This is a consequence of symmetry, as neither team has an advantage, thereby prolonging the competition.

Absorbing States in Markov Processes

The series of games can be modeled as a Markov process with absorbing states, where the process terminates once a team reaches the required number of wins. This framework uses state transitions and the property that once an absorbing state is reached, no further transitions occur.

Expected Value

This is the concept of computing a weighted average outcome of a random process. In problems involving games or repeated trials, the expected number of trials is calculated by summing over all possible outcomes weighted by their probabilities. It is a fundamental idea in probability theory used to predict the average result of random events.

Stopping Times

A stopping time is a random variable that represents the time at which a given stochastic process satisfies a specified condition. In this context, the series of games stops when one team wins a predetermined number of games, and the stopping time is the random number of games played until this condition is met.

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