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Chapter 5: Problem 64

According to Case Study \(4-2\) in Chapter 4 , the probability that a baseball player will have no hits in 10 trips to the plate is \(.0563\), given that this player has a batting average of \(.250\). Using the binomial formula, show that this probability is indeed \(.0563 .\)

Short Answer

Expert verified
The calculated probability using the binomial formula is indeed 0.0563, thus confirming that the probability of a baseball player with a batting average of .250 getting no hits in 10 trips to the plate is .0563.

Step by step solution

01

Identify given variables

The number of trials \(n\) is given as 10, which is the number of times the baseball player goes to the plate. The success probability per trial \(p\) is given as .250, that is the batting average of the player. It is necessary to find the probability that \(k\) equals 0 that is he will have no successful hits in 10 trials.
02

Substitution into the binomial formula

Substitute the known values into the formula. The binomial coefficient \(C(n, k) = C(10, 0)\) is actually equal to 1 (any number to the power of 0 is 1). So we are left with \((0.250)^0 \cdot (1 - 0.250)^{10}\).
03

Calculation

Calculate the probability: \(P(X = 0) = 1 \cdot 1 \cdot (0.75)^{10} = 0.0563\).
04

Verification

This value of probability matches with the given probability of 0.0563 in the problem. So, the calculated probability is indeed correct. So, the probability that a baseball player with a batting average of .250 gets no hits in 10 trips to the plate is indeed \(.0563\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Batting Average
The term "batting average" is a statistic used in baseball that helps measure a player's hitting performance. It's calculated by dividing the number of hits by the number of at-bats a player has. This number gives us a sense of how often the player successfully hits the ball in an official play.
For example, if a player has a batting average of 0.250, such as in our case study, it means that this player hits the ball 25% of the time he steps up to the plate. In simpler terms, out of every four times at bat, you can expect the player to get a hit roughly once. This average provides useful insight into the player's performance over a season.
Steps of Probability Calculation
Probability calculation allows us to determine the likelihood of certain outcomes happening. In our example, we're calculating the probability that a player with a batting average of 0.250 gets no hits in 10 tries at the plate.
This problem utilizes the binomial probability formula, where we're interested in the event of zero hits (k = 0). Let's summarize the key steps:
  • Identify your variables, namely the number of total attempts (n = 10) and the probability of success in each attempt (p = 0.250).
  • Use the binomial probability formula to calculate the chance of zero successful outcomes.
  • Substitute the values into the formula and simplify the equation.
  • Perform the calculation to find the probability. In this problem, it results in 0.0563, just like what's provided in the exercise.
Each of these steps is a fundamental building block in probability theory and can be applied to various statistical problems.
Role of the Binomial Coefficient
The binomial coefficient is a crucial part of the binomial probability formula and is often represented as \(C(n, k)\). It tells us how many ways we can pick \(k\) successes from \(n\) trials. The coefficient is a fundamental part of combinations and permutations, where order does not matter.
In our exercise, \(C(10, 0)\) indicates we're looking for ways to have zero successes out of ten tries, which is precisely one way: having no successful hit at all. So, \(C(10, 0) = 1\). Simplifying our calculations is much easier with this understanding, as it helps us focus just on calculating the probability when no successes occur.
Basics of Probability Theory
Probability theory is the mathematical framework for quantifying uncertainty. It deals with analyzing random events and making predictions about the likelihood of various outcomes.
Through probability theory, we address questions such as how likely it is for a player to score a hit. The theory relies on several key principles:
  • The probability of any event ranges from 0 (impossible) to 1 (certain).
  • Probabilities of all possible outcomes of a trial sum up to 1.
  • The application of probability distributions and formulas, like the binomial distribution used in our example.
Mastering these foundational elements of probability theory can help you approach many statistical problems methodically and predict potential outcomes accurately.

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Most popular questions from this chapter

Two teams, \(\mathrm{A}\) and \(\mathrm{B}\), will play a best-of-seven series, which will end as soon as one of the teams wins four games. Thus, the series may end in four, five, six, or seven games. Assume that each team has an equal chance of winning each game and that all games are independent of one another. Find the following probabilities. a. Team A wins the series in four games. b. Team A wins the series in five games. c. Seven games are required for a team to win the series.

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