Problem 92 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that two evenly matched teams are playing in the World Series. On the average, how many games will be played? (The winner is the first team to get four victories.) Assume that each game is an independent event.

Step-by-Step Solution

Answer

On average, 9 games will be played in the World Series.

Step 1: Identify the Distribution

This is a typical Negative Binomial Distribution problem. The Negative Binomial Distribution describes the probability of having to wait a certain amount of Bernoulli trials until achieving r successes, the parameters are (r,p), where p is the success probability for each trial and r = 4, indicating 4 wins to end the series. Since the two teams are evenly matched, the success probability \( p = 0.5 \).

Step 2: Calculate the Expected Value

The expected value (mean) for a Negative Binomial Distribution is given by \( E(X) = \frac{r}{p} \). In this case, the mean or expected number of games that will be played before a team wins four games is \( E(X) = \frac{r}{p} = \frac{4}{0.5} = 8 \) games.

Step 3: Adjust for Zero-Indexed Victory

However, it is important to note that once a team has won four games, the series is over, so we must add in the winning game. Hence the total number of games is \( E(X) + 1 = 8 + 1 = 9 \) games.

Key Concepts

Expected Value

The expected value in probability and statistics is a vital concept that helps us understand what outcome we might "expect" on average from a random experiment. In simpler terms, it tells us the predicted average over many trials. For a Negative Binomial Distribution, which is used to model the number of trials until a desired number of successes occurs, the expected value can be calculated using a straightforward formula: \[ E(X) = \frac{r}{p} \]where:

• r represents the number of successful outcomes required (in our case, 4 wins in the World Series),
• p is the probability of success in each trial (here, 0.5, indicating evenly matched teams).

To find out how many games are expected, plug the values into the formula: \[ E(X) = \frac{4}{0.5} = 8 \]This formula shows that, on average, 8 games will be needed. However, as the deciding game also counts, the series will typically end in about 9 games. This expected value helps to plan and anticipate the duration of such series.

Bernoulli Trials

Named after the mathematician Jacob Bernoulli, Bernoulli trials are basic experiments used in probability. Each trial has only two possible outcomes: success or failure. In the context of our World Series problem, every game played between the two teams is a Bernoulli trial:

• A success means one of the teams wins the game.
• A failure means the same team hasn't won yet.

Since the World Series involves playing multiple games until one team achieves 4 victories, these games are considered independent Bernoulli trials. The probability of success (team winning) for each trial in this problem is constant at 0.5 because both teams are equally matched. This concept of Bernoulli trials underpins the choice of using the Negative Binomial Distribution, as we want to determine the number of games required for a particular number of successes.

World Series Probability Problem

The World Series problem is a common probability problem where we calculate how many games will typically be played in a best-of-seven series between two evenly matched teams. The problem involves identifying the probability distribution that best describes the scenario and then determining how many trials (games) will likely occur before the decisive number of successes (wins). First, to tackle this, notice that each game's result can be treated as a Bernoulli trial, where each team has an equal chance of winning a single game, indicated by the probability 0.5. Then, the overall problem becomes one of applying the Negative Binomial Distribution. While the calculated expected number of games from the distribution was 8, the correct interpretation accounts for completing the series with a fourth win. That adjustment takes us to 9 games. This calculation reveals that in simulation and planning for similar tournaments, stakeholders might typically prepare for a series stretching to 9 games to allow for the actual conclusion of the series when a team achieves the 4th victory.