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Chapter 5: Problem 27

What are the parameters of the binomial probability distribution, and what do they mean?

Short Answer

Expert verified
The parameters of a binomial probability distribution are \(n\) and \(p\). \(n\) is the number of trials and \(p\) is the probability of success on a single trial.

Step by step solution

01

Identify the parameters

The binomial probability distribution has two parameters: \(n\) and \(p\).
02

Describe 'n' parameter

\'n\' represents the number of trials in the binomial experiment. This refers to how many times the experiment is performed.
03

Describe 'p' parameter

\'p\' represents the probability of success on a single trial. This means the likelihood of getting a successful outcome in one single trial.

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Key Concepts

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

Binomial Experiment
A binomial experiment is a statistical experiment that consists of a fixed number of independent trials, each with two possible outcomes. These outcomes are typically labeled as "success" and "failure." This type of experiment is foundational to understanding the binomial probability distribution.
  • Each trial is independent, meaning the result of one trial does not affect the others.
  • There are only two possible outcomes鈥攕uccess or failure鈥� in each trial.
  • The probability of success remains constant in every trial.
Binomial experiments are used in various fields, such as medicine for clinical trials, or in quality control processes in manufacturing.
Number of Trials
In a binomial experiment, the number of trials, represented by the parameter \( n \), is crucial. This term indicates how many times you perform the experiment.For instance, if you are flipping a coin 10 times, then \( n = 10 \). It is important that \( n \) remains constant for the study of a binomial distribution.
  • The more trials you conduct, the closer your results will align with theoretical probabilities.
  • Increasing the number of trials can reduce the variability in the results.
Probability of Success
The probability of success, denoted by \( p \), refers to the chance of achieving a successful result in a single trial. A success does not always mean something good; it simply refers to the event being measured, such as getting a 'head' in a coin toss.
  • This probability remains constant across all trials in a binomial experiment.
  • If you know \( p \), you can analyze and predict the probable outcomes using the binomial probability formula.
For example, if you roll a fair six-sided die and want the probability of rolling a 3 (our 'success'), then \( p = \frac{1}{6} \). This consistency allows statisticians to create models predicting outcomes of repeated experiments.
Statistical Parameters
In the context of a binomial distribution, the statistical parameters \( n \) and \( p \) provide the framework for calculating probabilities and analyzing outcomes.
  • These parameters allow you to determine the probability of achieving a certain number of successes in your trials.
  • They are essential for calculating the mean and variance of the distribution, which provide insights into the expected outcome and variability of results.
The mean of a binomial distribution is given by \( np \), and the variance is \( np(1-p) \). These expressions encapsulate the essence of the binomial distribution and guide statistical analysis and prediction of outcomes.

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Most popular questions from this chapter

Spoke Weaving Corporation has eight weaving machines of the same kind and of the same age. The probability is \(.04\) that any weaving machine will break down at any time. Find the probability that at any given time a. all eight weaving machines will be broken down b. exactly two weaving machines will be broken down c. none of the weaving machines will be broken down

Let \(x\) be the number of cars that a randomly selected auto mechanic repairs on a given day. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|ccccc} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .05 & .22 & .40 & .23 & .10 \\ \hline \end{array} $$ Find the mean and standard deviation of \(x\). Give a brief interpretation of the value of the mean.

Explain the hypergeometric probability distribution. Under what conditions is this probability distribution applied to find the probability of a discrete random variable \(x\) ? Give one example of an application of the hypergeometric probability distribution.

An instant lottery ticket costs \(\$ 2\). Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of \(\$ 5\) each, 100 tickets have a prize of \(\$ 10\) each, 5 tickets have a prize of \(\$ 1000\) each, and 1 ticket has a prize of \(\$ 5000\). Let \(x\) be the random variable that denotes the net amount a player wins by playing this lottery. Write the probability distribution of \(x\). Determine the mean and standard deviation of \(x\). How will you interpret the values of the mean and standard deviation of \(x\) ?

Although Borok's Electronics Company has no openings, it still receives an average of \(3.2\) unsolicited applications per week from people seeking jobs. a. Using the Poisson formula, find the probability that this company will receive no applications next week. b. Let \(x\) denote the number of applications this company will receive during a given week. Using the Poisson probabilities table from Appendix B, write the probability distribution table of \(x\). c. Find the mean, variance, and standard deviation of the probability distribution developed in part b.
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