Problem 52 - Basic Statistics for Business and Economics with Student

Question

Joe Mauer of the Minnesota Twins had the highest batting average in the 2006 Major League Baseball season. His average was . \(347 .\) So assume the probability of getting a hit is . 347 for each time he batted. In a particular game assume he batted three times. a. This is an example of what type of probability? b. What is the probability of getting three hits in a particular game? c. What is the probability of not getting any hits in a game? d. What is the probability of getting at least one hit?

Step-by-Step Solution

Answer

a. Binomial probability. b. 0.0417 c. 0.278 d. 0.722

Step 1: Identify the probability type

The problem describes a scenario where the probability of a hit is consistent each time Joe bats. This is an example of a binomial probability because each time he bats, it's an independent trial with two possible outcomes (a hit or not a hit).

Step 2: Formulate for three hits

The probability of Joe getting exactly 3 hits in 3 at-bats can be calculated using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(n = 3\), \(k = 3\), and \(p = 0.347\).\[ P(X = 3) = \binom{3}{3} (0.347)^3 (1-0.347)^{3-3} \]\[ P(X = 3) = 1 \times (0.347)^3 \times (0.653)^0 \]\[ P(X = 3) = 0.0417 \]

Step 3: Calculate probability of zero hits

Using the binomial probability formula again, but this time \(k = 0\):\[ P(X = 0) = \binom{3}{0} (0.347)^0 (1-0.347)^3 \]\[ P(X = 0) = 1 \times 1 \times 0.653^3 \]\[ P(X = 0) = 0.278 \]

Step 4: Calculate probability of at least one hit

To find the probability of getting at least one hit, we can use the complement rule and subtract the probability of getting zero hits from 1:\[ P(X \geq 1) = 1 - P(X = 0) \]\[ P(X \geq 1) = 1 - 0.278 \]\[ P(X \geq 1) = 0.722 \]

Key Concepts

Understanding Batting Average

The batting average is a crucial statistic in baseball, representing a player's hitting performance. It is calculated by dividing the number of hits a player makes by the total number of at-bats. For example, Joe Mauer had a batting average of .347 in 2006, which means he got a hit approximately 34.7% of the time he batted.
This value reflects the probability that Joe would get a hit on any given at-bat during that season. In probability terms, a batting average is a straightforward way to express the likelihood of success in any given attempt. In Joe's case, every time he stepped up to bat, he had a .347 probability of getting a hit.
Batting averages provide a simple and understandable method for analyzing player performance and making predictions about future plays.

The Role of Independent Trials

In probability theory, independent trials are scenarios where the outcome of one event does not affect the outcome of another. This is a fundamental concept when determining probabilities in sports and other real-life applications.
Joe Mauer's at-bats can be seen as independent trials because each of his attempts is separate from the others. No matter how he performed in previous at-bats, his chance of hitting in the next remains the same, at .347.
Understanding independent trials is essential in calculating probabilities accurately, as it confirms that each trial is influenced by the same initial conditions.

Probability Calculation in Action

When it comes to calculating the probability of a specific outcome, such as the number of hits Joe Mauer might achieve, the process relies heavily on established statistical formulas. One commonly used formula is the binomial probability formula:

• \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

In this equation, \(n\) represents the total number of trials (in Joe's case, at-bats), \(k\) is the number of successful trials (hits), and \(p\) is the probability of success on a single trial.
This formula helps determine the probability of achieving exactly \(k\) successes in \(n\) trials, facilitating a deeper understanding of potential outcomes. For example, if Joe batted three times, you can use this formula to find the probability of getting exactly three hits.

Understanding Binomial Distribution

The binomial distribution is a specific probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials. Each trial is a binary event, meaning it has only two possible outcomes.
In the context of baseball, considering Joe Mauer's at-bats, each batting instance can result in a hit (success) or no hit (failure).The distribution is characterized by parameters like the number of trials, \(n\), and the probability of success, \(p\).
Utilizing the binomial distribution allows us to model and calculate probabilities like the likelihood of Joe Mauer achieving a certain number of hits during a game. This statistical tool is invaluable in many fields beyond sports, offering a framework for analyzing events with defined success-failure outcomes.