Problem 8 - Algebra 2

Question

Lauren Wible of Bucknell University was the 2005 NCAA Division I women's softball batting leader with a batting average of .524. This means that the probability of her getting a hit in a given at-bat was \(0.524 .\) Find the probability of her getting exactly 2 hits in 4 at-bats.

Step-by-Step Solution

Answer

The probability is approximately 0.3753.

Step 1: Understand the Problem

Lauren Wible's batting average is 0.524, which means the probability of her getting a hit in a single at-bat is 0.524. We need to find the probability of her getting exactly 2 hits in 4 at-bats.

Step 2: Identify the Distribution

This problem uses the binomial distribution because it involves a fixed number of independent trials (at-bats), and each trial has two possible outcomes (hit or no hit) with a constant probability of success (hitting).

Step 3: Set the Parameters of the Binomial Distribution

For a binomial distribution, we'll define the parameters as: - Number of trials ( =n ) = 4 - Probability of success in each trial ( p ) = 0.524 - Number of successes ( k ) we are interested in = 2.

Step 4: Apply the Binomial Probability Formula

The probability of getting exactly kgiven successes in n trials is given by the formula for the binomial probability: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substitute the values: \[ P(X = 2) = \binom{4}{2} (0.524)^2 (1-0.524)^{4-2} \]

Step 5: Calculate the Binomial Coefficient

Compute the binomial coefficient \(\binom{4}{2}\): \[\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

Step 6: Calculate the Probability

Compute the individual components and substitute them into the formula: - \( (0.524)^2 = 0.274576 \)- \((1-0.524)^{2} = (0.476)^2 = 0.226576 \) Then calculate the probability: \[ P(X=2) = 6 \times 0.274576 \times 0.226576 \approx 0.3753 \]

Step 7: Conclusion

The probability of Lauren Wible getting exactly 2 hits in 4 at-bats is approximately 0.3753.

Key Concepts

Probability

Probability is a fundamental concept in statistics that helps us quantify the likelihood of different outcomes. In this context, probability refers to the chance that a particular event occurs. For Lauren Wible, her batting average translates directly into the probability that she will hit the ball during an at-bat.

• Probability of success (getting a hit): 0.524
• Probability of failure (missing a hit): 1 - 0.524 = 0.476

The concept lies at the heart of predicting outcomes across different trials. Whether in sports or other fields, understanding probability allows us to anticipate possible results given a set of conditions.

Batting Average

A batting average is a key performance metric in baseball and softball. It represents the ratio of a player's hits compared to their total number of at-bats. For Lauren Wible, a batting average of 0.524 means that she gets a hit approximately 52.4% of the time she is at-bat.

• Calculated as: \[\text{Batting Average} = \frac{\text{Number of Hits}}{\text{Number of At-Bats}}\]
• Helps in predicting future performance and assessing consistency.

This statistic not only indicates the player's skill level but also feeds into statistical models like the binomial distribution to predict outcomes over multiple trials.

Binomial Coefficient

The binomial coefficient is a critical component in calculating probabilities in a binomial distribution. It tells us the number of ways we can choose a certain number of successes in a series of trials.

• Mathematically expressed using combinations, for example: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!}\]
• For Lauren's scenario: \[ \binom{4}{2} = \frac{4!}{2!\times(4-2)!} = 6\]
• Represents the different ways she can get exactly 2 hits in 4 at-bats.

Understanding the binomial coefficient helps us analyze the structure of outcomes—in this case, predicting how likely multiple specific outcomes are.

Independent Trials

Independent trials refer to situations where the outcome of one trial does not affect the outcome of another. For Lauren Wible's batting outcomes:

• Each at-bat is an independent event.
• The probability of getting a hit remains constant at 0.524, no matter the result of previous at-bats.

The assumption of independence is crucial in applying the binomial distribution. It allows for the simplification of calculations. Knowing that each swing of the bat is independent from the next, statisticians can easily model and predict sequences of outcomes over time using probability distributions.