/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 In a binomial experiment, is it ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

In a binomial experiment, is it possible for the probability of success to change from one trial to the next? Explain.

Short Answer

Expert verified
No, the probability of success cannot change in a binomial experiment; it must remain constant across trials.

Step by step solution

01

Understanding Binomial Experiments

A binomial experiment is a statistical experiment that meets the following criteria: 1) The experiment consists of a fixed number of trials, 2) Each trial is independent, 3) Each trial has only two possible outcomes: success or failure, 4) The probability of success is the same for each trial.
02

Fixed Probability of Success

In a binomial experiment, one of its defining characteristics is that the probability of success remains constant across all trials. This means that the likelihood of achieving a success in any trial is not influenced by previous trials or future trials; it is fixed for every single trial of the experiment.
03

Conclusion on Changing Probability

Given that a binomial experiment requires the probability of success to remain the same for every trial, it is not possible for the probability of success to change from one trial to the next. If the probability changes, the experiment would not be considered a binomial experiment.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability of Success
In a binomial experiment, the probability of success is a crucial component. It represents the likelihood that any single trial results in what is defined as a success. For the experiment to be classified as a binomial, this probability must remain constant throughout all trials.

This consistency in probability is fundamental. It ensures that each trial behaves predictably and independently.
  • For example, if you're flipping a fair coin, the probability of landing heads (considered a success) is 0.5 for each flip.
  • This probability does not change regardless of how many times the coin is flipped before or after.
The constancy of the probability of success differentiates a binomial experiment from other types of statistical experiments where probabilities might vary.
Independent Trials
A defining feature of a binomial experiment is that all trials must be independent. This means the outcome of one trial must have no influence over the outcome of another.

Independence is a cornerstone of a binomial framework. It allows each trial to stand alone, unaffected by prior or subsequent trials, ensuring that the results are purely based on chance rather than influence from other factors.
  • Imagine rolling a dice; whether or not you roll a six on one throw doesn鈥檛 affect what you might roll next.
  • The trials are isolated events, adding to the reliability of the experiment's results.
Independent trials help maintain the integrity of a binomial experiment by ensuring results are not skewed or biased by other outcomes.
Fixed Number of Trials
In a typical binomial experiment, the number of trials is predetermined and remains unchanged. This fixed number is an integral aspect, used to define the scope and potential outcomes of the experiment.

The predetermined number of trials offers a clear framework within which the experiment operates, ensuring consistency.
  • Consider a scenario where you flip a coin 10 times; the number of flips doesn't change once the experiment starts.
  • This fixed nature allows for structured data collection and simplified analysis.
By defining a specific number of trials in advance, the experimenter can plan effectively and analyze results confidently.
Binary Outcomes
Every trial within a binomial experiment is characterized by one of two possible outcomes, often labeled as "success" or "failure." These binary outcomes simplify the decision-making and analysis process.

Having only two possible results per trial ensures clarity and straightforwardness in the interpretation of data.
  • For example, when shooting a basketball, a successful shot is a "success," and a miss is a "failure."
  • There are no middle-ground outcomes; a shot either goes in or it does not.
This binary nature aids in establishing clear expectations and simplifies calculations, as outcomes are limited to just two possibilities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The one-time fling! Have you ever purchased an article of clothing (dress, sports jacket, etc.), worn the item once to a party, and then returned the purchase? This is called a one-time fling. About \(10 \%\) of all adults deliberately do a one-time fling and feel no guilt about it (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press). In a group of seven adult friends, what is the probability that (a) no one has done a one-time fling? (b) at least one person has done a one-time fling? (c) no more than two people have done a one-time fling?

Which of the following are continuous variables, and which are discrete? (a) Number of traffic fatalities per year in the state of Florida (b) Distance a golf ball travels after being hit with a driver (c) Time required to drive from home to college on any given day (d) Number of ships in Pearl Harbor on any given day (e) Your weight before breakfast each morning

Aldrich Ames is a convicted traitor who leaked American secrets to a foreign power. Yet Ames took routine lie detector tests and each time passed them. How can this be done? Recognizing control questions, employing unusual breathing patterns, biting one's tongue at the right time, pressing one's toes hard to the floor, and counting backward by 7 are countermeasures that are difficult to detect but can change the results of a polygraph examination (Source: Lies' Liesi't Lies\%' The Psychology of Deceit, by C. V. Ford, professor of psychiatry, University of Alabama). In fact, it is reported in Professor Ford's book that after only 20 minutes of instruction by "Buzz" Fay (a prison inmate), \(85 \%\) of those trained were able to pass the polygraph examination even when guilty of a crime. Suppose that a random sample of nine students (in a psychology laboratory) are told a "secret" and then given instructions on how to pass the polygraph examination without revealing their knowledge of the secret. What is the probability that (a) all the students are able to pass the polygraph examination? (b) more than half the students are able to pass the polygraph examination? (c) no more than four of the students are able to pass the polygraph examination? (d) all the students fail the polygraph examination?

Approximately \(3.6 \%\) of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Source: Australian Journal of Agricultural Research, Vol. 25, pp. 783-790). (Bitter pit is a disease of apples resulting in a soggy core, which can be caused either by over-watering the apple tree or by a calcium deficiency in the soil.) Let \(n\) be a random variable that represents the first Jonathan apple chosen at random that has bitter pit. (a) Write out a formula for the probability distribution of the random variable \(n\) (b) Find the probabilities that \(n=3, n=5,\) and \(n=12\) (c) Find the probability that \(n \geq 5\) (d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use \(\mu\) for the geometric distribution and round.

USA Today reported that about 20\% of all people in the United States are illiterate. Suppose you interview seven people at random off a city street. (a) Make a histogram showing the probability distribution of the number of illiterate people out of the seven people in the sample. (b) Find the mean and standard deviation of this probability distribution. Find the expected number of people in this sample who are illiterate. (c) How many people would you need to interview to be 98\% sure that at least seven of these people can read and write (are not illiterate)?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.