Question

Teams \(A\) and \(B\) go into sudden death overtime after playing to a tie. The teams alternate possession of the ball and the first team to score wins. Each team has a \(\frac{1}{6}\) chance of scoring when it has the ball, with Team \(\mathrm{A}\) having the ball first. a. The probability that Team A ultimately wins is \(\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}\) Evaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is \(\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1} .\) Evaluate this series.

Short answer

Answer: The probability that Team A ultimately wins is 6/11, and the expected number of rounds required for the overtime to end is 6.

Step-by-step solution

(Compute the sum of the infinite geometric series)

First, let's tackle part a. The probability that Team A ultimately wins is given by the infinite geometric series: $$\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}$$ The general term of the series is: $$a_k = \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}$$ Since the common ratio of the geometric series is \(\left(\frac{5}{6}\right)^2\), and its absolute value is less than 1, we can use the formula for the sum of an infinite geometric series: $$S = \frac{a_0}{1 - r}$$ where \(a_0\) is the first term and \(r\) is the common ratio.

(Calculate Team A's winning probability)

Now let's calculate the probability that Team A ultimately wins: $$ S = \frac{\frac{1}{6}}{1 - \left(\frac{5}{6}\right)^2} = \frac{\frac{1}{6}}{1 - \frac{25}{36}} = \frac{\frac{1}{6}}{\frac{11}{36}} = \frac{6}{11} $$ The probability that Team A ultimately wins is \(\frac{6}{11}\).

(Evaluate the expected number of rounds)

Next, we need to evaluate the summation for part b. The expected number of rounds is given by the summation: $$\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1}$$ To find the sum of this summation, we can use the following formula for the sum of an infinite geometric series with factor \(k\): $$\sum_{k=1}^{\infty} kr^k = \frac{1}{(1-r)^2}$$ Considering \(r = \frac{5}{6}\), we can apply the above formula.

(Calculate the expected number of rounds)

We will now calculate the expected number of rounds required for the overtime to end. Multiplying the summation by \(\frac{1}{6}\), we have: $$\frac{1}{6} \cdot \frac{1}{\left(1 - \frac{5}{6}\right)^2} = \frac{1}{6} \cdot \frac{1}{\left(\frac{1}{6}\right)^2} = 6$$ The expected number of rounds required for the overtime to end is 6. In conclusion, the probability that Team A ultimately wins is \(\frac{6}{11}\) and the expected number of rounds required for the overtime to end is 6.

Key concepts

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a constant called the common ratio. It appears frequently in probability and statistics. Understanding geometric series helps us model scenarios where outcomes are based on repeated trials, each with a probabilistically independent chance of occurring.

• The series is written as \(a, ar, ar^2, ar^3, \ldots\)
• The sum of an infinite geometric series is calculated using \(S = \frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio.
• If the absolute value of \(r\) is less than 1, the series converges, meaning it approaches a finite sum.

Expectation

In probability, expectation refers to the average value or mean of a random variable based on its probability distribution. The expected value essentially provides a measure of the center of the distribution of the variable. It's like asking, "What result should I expect on average, over many trials?"

• For a discrete random variable \(X\), the expected value \( E(X) \) is the sum of possible values multiplied by their probabilities: \( E(X) = \sum x_i P(x_i) \).
• In scenarios involving repeated processes, like tossing a coin or sudden death overtime scenarios, expectation helps predict the average number of moves or plays needed.

Expectation in Series

When evaluating expected values related to series, such as those involving infinite geometric terms, specialized formulas are used. For instance, if you're dealing with an infinite geometric series multiplied by \(k\), then \(\sum_{k=1}^{\infty} kr^k = \frac{1}{(1-r)^2}\). This formula allows calculations of expected times or frequencies.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Understanding infinite series is crucial because they allow mathematicians and scientists to model continuous processes with infinite steps.

• Just like geometric series, infinite series can sometimes converge, meaning they reach a finite sum, which is useful in practical applications.
• An infinite series \(\sum_{k=0}^{\infty} a_k\) converges when the terms \(a_k\) approach zero as \(k\) increases and the partial sums approach a finite number.
• Convergence criteria, like the common ratio in geometric series being less than 1, help determine if an infinite series will converge.

Probability Calculation

Probability calculation involves methods and techniques used to determine the likelihood of certain outcomes. When expressed in fractions or percentages, probability efficiently quantifies how likely an event is to occur.

• Basic probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes.
• Probability calculations form the backbone of statistical and probabilistic analysis, useful in decision-making and predicting outcomes.
• More complex probability scenarios often involve formulas derived from geometric or infinite series, especially where outcomes relate to repeated trials like those in overtime strategies.

Application in Overtime Strategies

In the sudden death overtime problem, probability helps to determine important scenarios, such as which team is more likely to win first. By understanding probability, we can derive expressions for win probability and expected plays based on the given scoring chances with calculations involving geometric series.