Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.

a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

검증된 단계별 안내

Recognize that the given series is an infinite geometric series of the form \(\sum_{k=0}^\infty ar^k\), where the first term \(a = \frac{1}{6}\) and the common ratio \(r = \left(\frac{5}{6}\right)^2\).

Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\).

Substitute the values of \(a\) and \(r\) into the formula: \(S = \frac{\frac{1}{6}}{1 - \left(\frac{5}{6}\right)^2}\).

Simplify the denominator by calculating \(\left(\frac{5}{6}\right)^2 = \frac{25}{36}\), so the denominator becomes \(1 - \frac{25}{36}\).

Express the denominator as a single fraction and then write the sum \(S\) as a simplified fraction or decimal (without calculating the final numeric value).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. It has the form ∑ ar^k, and if |r| < 1, the series converges to a/(1-r). This concept is essential for evaluating infinite sums like the one given in the problem.

Probability of Independent Events

When events are independent, the probability of their joint occurrence is the product of their individual probabilities. Here, each scoring attempt is independent, so the probability of sequences of successes and failures can be calculated by multiplying the respective probabilities.

Infinite Sum in Probability Context

In problems involving repeated trials until success, probabilities often form infinite sums representing all possible outcomes. Understanding how to express and evaluate these sums helps determine the total probability of an event occurring eventually, such as Team A winning in sudden-death overtime.