/*! 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 21 Writing Explain how a binomial e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 21

Writing Explain how a binomial experiment is related to a binomial expansion.

Short Answer

Expert verified
A binomial experiment, where there are n trials and two possible outcomes, success (p) and failure (q), is related to a binomial expansion in that the probability of getting exactly k successes in the experiment can be given by a term in the binomial expansion, specifically \( \binom{n}{k}(p^{k})(q^{n-k}) \)

Step by step solution

01

Define a Binomial Experiment

A binomial experiment is one that possesses the following properties: \n-The experiment consists of n repeated trials. \n-Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. \n-The probability of success, denoted by P, is the same on every trial. \n-The trials are independent; the outcome on one trial does not affect the outcome on other trials.
02

Define a Binomial Expansion

A binomial expansion refers to the algebraic expansion of powers of a binomial. According to the Binomial theorem, it can be expanded as the sum of n+1 terms of the form: \( \binom{n}{k}(a^{n-k})(b^k) \) for k = 0, 1, 2, ..., n where \( \binom{n}{k} \) are the binomial coefficients.
03

Relate Binomial Experiment to Binomial Expansion

In a binomial experiment with n trials and probability of success p; the probability of getting exactly k successes is given by the term \( \binom{n}{k}(p^{k})(q^{n-k}) \) , where q = 1 - p. This term is similar to a term in the binomial expansion. Hence, we see that the binomial expansion gives us the different probabilities associated with 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.

Understanding Binomial Experiments
A binomial experiment is an important concept in probability theory. You can think of it as a series of trials designed with certain conditions in mind. Let's break it down.
  • Fixed Number of Trials: The experiment involves a specific number of trials, denoted by \(n\). Whether it's flipping a coin or drawing cards, the number of trials remains constant.

  • Two Possible Outcomes: Each trial concludes with one of two possibilities鈥攕uccess or failure. It's like answering a true or false question, where success is one answer, and failure is another.

  • Constant Probability: The probability \(P\) that a trial will succeed stays the same across all trials. If you're flipping a fair coin, that probability remains 0.5 each time.

  • Independence of Trials: The outcomes of trials are independent, meaning the result of one trial does not affect another. If the first flip is heads, it won鈥檛 affect the result of the following flips.
This setup allows the binomial experiment to model various real-world phenomena where you're interested in the number of successful outcomes over several attempts.
Exploring Binomial Expansion
Binomial expansion stems from mathematics and provides a way to expand expressions raised to powers. But what exactly does this mean?
  • Expression Structure: A binomial expression is a sum consisting of two terms, like \((a + b)\). When you raise it to a power, such as \((a + b)^n\), you need to expand it out.

  • Theorem Application: The Binomial Theorem is the tool used for this expansion. It tells us that \((a + b)^n\) can be written as a sum of several terms, specifically \(n+1\) terms.

  • Terms of Expansion: Each term takes the form \( \binom{n}{k} (a^{n-k})(b^k) \). Here \( \binom{n}{k} \) represents the binomial coefficients, which are numerical values that depend on \(n\) and \(k\).
This expansion is fundamental in algebra and finds applications in various fields, linking directly to how probabilities are calculated in a binomial experiment.
Connecting Probability Theory with Binomial Concepts
Probability theory provides the framework for predicting how likely events are to occur. When we connect binomial experiments to binomial expansions, this framework becomes very handy.
  • Calculating Exact Successes: In a binomial experiment, you might want to discover the probability of exactly \(k\) successes. Here, probability theory helps by using the formula \(\binom{n}{k} (p^k)(q^{n-k})\), where \(q = 1 - p\).

  • Role of Binomial Coefficients: The \(\binom{n}{k}\) component is a binomial coefficient, mirroring its role in binomial expansion. It helps in determining possible ways to achieve \(k\) successes.

  • Understanding Outcomes: This probability calculation is akin to selecting a specific term from the binomial expansion. Therefore, the expansion gives us a comprehensive view of all possible outcomes and their probabilities in a binomial setup.
Thus, understanding the connection between binomial expansion and binomial experiments allows for a deeper insight into applied probability theory.

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

Quality Control Tubs of Better Butter weigh 1.0 each, with a standard deviation of 0.06 Ib. At a quality control checkpoint, 12 of the tubs taken as samples weighed less than 0.88 b. Assume that the weights of the samples were normally distributed. How many tubs of butter were taken as samples?

Factor each expression. $$ x^{2}+6 x+9 $$

Factor each expression. $$ 3 x^{2}+12 x+9 $$

Four percent of the tenants in an apartment building live alone. Suppose five tenants are selected randomly. Which expression represents \(P(\text { all live alone alone)? }\) \(\begin{array}{llll}{\text { A. }(0.04)^{5}} & {\text { B. }(0.4)^{5}} & {\text { C. }(0.96)^{5}} & {\text { D. }(5)^{0.04}}\end{array}\)

Another measure of variation is variance, which equals \(\sigma^{2} .\) Find the variance and the standard deviation of each data set. $$ 60 \mathrm{m} 40 \mathrm{m} \quad 35 \mathrm{m} 45 \mathrm{m} 39 \mathrm{m} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.