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Chapter 5: Problem 28

This problem will be referred to in the study of control charts (Section 6.1). In the binomial probability distribution, let the number of trials be \(n=3,\) and let the probability of success be \(p=0.0228 .\) Use a calculator to compute (a) the probability of two successes. (b) the probability of three successes. (c) the probability of two or three successes.

Short Answer

Expert verified
(a) 0.00152, (b) 0.00001186, (c) 0.00153186.

Step by step solution

01

Identify Binomial Probability Formula

The probability of obtaining exactly \(k\) successes in \(n\) independent Bernoulli trials with probability \(p\) is given by the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient.
02

Calculate Probability of Two Successes

Substitute the values \(n = 3\), \(p = 0.0228\), and \(k = 2\) into the binomial formula: \[ P(X = 2) = \binom{3}{2} (0.0228)^2 (1-0.0228)^{3-2} \] Calculate \( \binom{3}{2} = 3 \) and then \(P(X=2) = 3 \times 0.0228^2 \times 0.9772 \). Finally, compute: \[ P(X=2) \approx 0.00152 \]
03

Calculate Probability of Three Successes

Substitute the values \(n = 3\), \(p = 0.0228\), and \(k = 3\) into the binomial formula: \[ P(X = 3) = \binom{3}{3} (0.0228)^3 (1-0.0228)^{3-3} \] Calculate \( \binom{3}{3} = 1 \) and then \(P(X=3) = 1 \times 0.0228^3 \). Finally, compute: \[ P(X=3) \approx 0.00001186 \]
04

Calculate Probability of Two or Three Successes

The probability of two or three successes, \(P(X = 2 \text{ or } X = 3)\), is the sum of the individual probabilities: \[ P(X = 2 \text{ or } X = 3) = P(X = 2) + P(X = 3) \]Substituting the previously calculated probabilities: \[ P(X = 2 \text{ or } X = 3) \approx 0.00152 + 0.00001186 \]Thus, \[ P(X = 2 \text{ or } X = 3) \approx 0.00153186 \]

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Key Concepts

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

Control Charts
Control charts are an essential tool in the field of quality control and process monitoring. They help in visualizing the performance of a process over time. By plotting data points in time order, you can determine whether a process is stable or if there are any irregularities indicating a potential issue.
  • They show the variation in a dataset.
  • Highlight any trends or outliers.
  • Are extensively used in manufacturing and other industries to ensure consistent quality.
These charts often leverage statistical measures like the binomial probability distribution to determine if changes seen in the chart are random or due to specific reasons. This allows businesses to make informed decisions about process modifications and improvement.
Binomial Coefficient
The binomial coefficient, denoted as \( \binom{n}{k} \), is a fundamental concept in the calculation of binomial probabilities. It determines the number of ways to choose \(k\) successes from \(n\) trials and is calculated using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
  • The symbol \(!\) denotes a factorial, which is the product of all positive integers up to a specified number.
  • The binomial coefficient is critical when determining probabilities in situations involving multiple trials, such as flipping a coin multiple times or predicting the likelihood of a certain number of successes in Bernoulli trials.
Understanding the binomial coefficient helps in not only calculating probabilities but also in comprehending combinations and permutations in broader mathematical contexts.
Probability of Success
The probability of success, symbolized as \(p\), is a key parameter in any binomial probability distribution. It represents the likelihood of a desired outcome occurring in a single trial.
  • Each trial in a binomial setting is an independent Bernoulli trial, which can result in either a success or a failure.
  • In the exercise example, \(p = 0.0228\), meaning there is approximately a 2.28% chance of achieving success in a single trial.
Grasping the concept of the probability of success helps you in making accurate predictions regarding the outcome of experiments and understanding how changes in probability affect overall results.
Bernoulli Trials
Bernoulli trials form the basis of the binomial probability distribution. Each trial can have one of two possible outcomes: success or failure. Some key features of Bernoulli trials include:
  • Each trial is independent, meaning the outcome of one trial does not affect the outcome of another.
  • The probability of success remains constant for each trial.
When we discuss a series of Bernoulli trials, we are generally referring to a situation involving repetition, such as multiple coin tosses or repeated quality checks. These trials are the building blocks for larger, more complex probability models and are crucial for applications where predicting a number of successes is essential.

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Most popular questions from this chapter

What does the expected value of a binomial distribution with \(n\) trials tell you?

Suppose we have a binomial experiment with 50 trials, and the probability of success on a single trial is 0.02. Is it appropriate to use the Poisson distribution to approximate the probability of two successes? Explain.

From long experience a landlord knows that the probability an apartment in a complex will not be rented is \(0.10 .\) There are 20 apartments in the complex, and the rental status of each apartment is independent of the status of the others. When a minimum of 16 apartment units are rented, the landlord can meet all monthly expenses. Which probability is more relevant to the landlord in terms of being able to meet expenses: the probability that there are exactly four unrented units or the probability that there are four or fewer unrented units? Explain.

The owners of a motel in Florida have noticed that in the long run, about \(40 \%\) of the people who stop and inquire about a room for the night actually rent a room. (a) How many inquiries must the owner answer to be \(99 \%\) sure of renting at least one room? (b) If 25 separate inquiries are made about rooms, what is the expected number of inquiries that will result in room rentals?

USA Today reported that the U.S. (annual) birthrate is about 16 per 1000 people, and the death rate is about 8 per 1000 people. (a) Explain why the Poisson probability distribution would be a good choice for the random variable \(r=\) number of births (or deaths) for a community of a given population size. (b) In a community of 1000 people, what is the (annual) probability of 10 births? What is the probability of 10 deaths? What is the probability of 16 births? 16 deaths? (c) Repeat part (b) for a community of 1500 people. You will need to use a calculator to compute \(P(10 \text { births) and } P(16\text { births). }\) (d) Repeat part (b) for a community of 750 people.
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