We are playing a tournament in which we stop as soon as one...

We are playing a tournament in which we stop as soon as one of us wins $n$ games. We are evenly matched, so each of us wins any game with probability $1 / 2$, independently of other games. What is the probability that the loser has won $k$ games when the match is over?

Key Concepts

Bernoulli Trials

Bernoulli trials refer to experiments or games that have exactly two possible outcomes (like win or loss) with a fixed probability for each outcome, and these outcomes are independent of each other. This set-up is key when modeling sequences of games, where each game represents an independent trial with a win probability of 1/2.

Stopping Criterion

The stopping criterion is a rule that terminates a process once a specific condition is met. In tournament problems, the process stops when one player reaches a predetermined number of wins. This concept is essential to limit the number of games played and directly impacts the way probabilities are calculated for outcomes preceding this termination.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways an event can occur, which is crucial when evaluating the different sequences in which wins and losses can happen. In tournaments, it enables the calculation of the sequences that lead to a match ending with a specific number of wins for the loser.

Negative Binomial Distribution

The negative binomial distribution models the number of failures (or in this context, the number of games won by the losing player) before achieving a specified number of successes (or wins by the winning player). It is useful for determining probabilities in processes that continue until a fixed number of successes occurs.

Symmetry in Probability

Symmetry in probability means that the probability structure remains the same regardless of which player is considered, owing to identical chances of winning. This concept simplifies analysis in balanced games or tournaments, as it allows for the inference that results for one player mirror those for the other.

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