/*! 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 23 A binomial probability experimen... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 23

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=9, p=0.2, x \leq 3 $$

Short Answer

Expert verified
\( P(X \leq 3) \)

Step by step solution

01

Understand the Binomial Distribution

For a binomial distribution, the probability of getting exactly x successes in n trials is given by the formula: \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \] where\( n \) is the number of trials, \( p \) is the probability of success, and \( \binom{n}{x} \) is the binomial coefficient.
02

Find Binomial Coefficients

Using the binomial coefficient formula, \( \binom{n}{x} = \frac{n!}{x!(n-x)!} \), we need to calculate the coefficients for each value of \( x \) from 0 to 3.
03

Calculate Individual Probabilities

Compute the probability of getting exactly 0, 1, 2, and 3 successes using the binomial probability formula: \[ P(X = 0) = \binom{9}{0} (0.2)^0 (0.8)^9 \] \[ P(X = 1) = \binom{9}{1} (0.2)^1 (0.8)^8 \] \[ P(X = 2) = \binom{9}{2} (0.2)^2 (0.8)^7 \] \[ P(X = 3) = \binom{9}{3} (0.2)^3 (0.8)^6 \]
04

Sum the Probabilities

Add up all the individual probabilities to find the total probability of getting at most 3 successes. \[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]

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.

Binomial Distribution
A binomial distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent trials. Each trial has two possible outcomes: success or failure. Here's a simple way to understand it:

The binomial distribution is defined by two parameters:
  • - **n**: the number of trials
  • - **p**: the probability of success in each trial
In our exercise, we have 9 trials, and the probability of success is 20%. This means we are dealing with a binomial distribution with parameters n=9 and p=0.2.

The general formula used to find the probability of exactly x successes in n trials is:

\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]

Don't worry, we'll break down each part of this formula in the next sections.
Binomial Coefficient
The binomial coefficient helps us understand how many ways we can choose x successes out of n total trials. It's represented as:

\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]

Here, the exclamation mark denotes a factorial, which is the product of all positive integers up to that number. For example, 4! = 4脳3脳2脳1 = 24.

It's important to calculate the binomial coefficients for each value of x (0 through 3) before moving to the probability calculation:
  • \( \binom{9}{0} = \frac{9!}{0!(9-0)!} = 1 \)
  • \( \binom{9}{1} = \frac{9!}{1!(9-1)!} = 9 \)
  • \( \binom{9}{2} = \frac{9!}{2!(9-2)!} = 36 \)
  • \( \binom{9}{3} = \frac{9!}{3!(9-3)!} = 84 \)
These coefficients will be used in the next step to compute the exact probabilities.
Probability Calculation
Now, we'll use the binomial probability formula to calculate the probability of getting exactly 0, 1, 2, and 3 successes:

  • For 0 successes:
    • \[ P(X = 0) = \binom{9}{0} (0.2)^0 (0.8)^9 \]\[ = 1 \times 1 \times 0.1342 = 0.1342 \]
  • For 1 success:
    • \[ P(X = 1) = \binom{9}{1} (0.2)^1 (0.8)^8 \]\[ = 9 \times 0.2 \times 0.2684 = 0.4822 \]
  • For 2 successes:
    • \[ P(X = 2) = \binom{9}{2} (0.2)^2 (0.8)^7 \]\[ = 36 \times 0.04 \times 0.5033 = 0.7253 \]
  • For 3 successes:
    • \[ P(X = 3) = \binom{9}{3} (0.2)^3 (0.8)^6 \]\[ = 84 \times 0.008 \times 0.7937 = 0.5302 \]
Finally, sum these probabilities to get the total probability of getting at most 3 successes:

\[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]\[ = 0.1342 + 0.4822 + 0.7253 + 0.5302 = 1.872 \]

Hence, the probability of getting at most 3 successes in 9 trials, each with a 20% success rate, is approximately 1.872.

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 expected number of successes in a binomial experiment with \(n\) trials and probability of success \(p\) is ______.

Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The number of defects in a roll of carpet. (b) The distance a baseball travels in the air after being hit. (c) The number of points scored during a basketball game. (d) The square footage of a house.

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. Three cards are selected from a standard 52 -card deck with replacement. The number of kings selected is recorded.

In the following probability distribution, the random variable \(X\) represents the number of marriages an individual aged 15 years or older has been involved in. $$ \begin{array}{ll} x & P(x) \\ \hline 0 & 0.272 \\ \hline 1 & 0.575 \\ \hline 2 & 0.121 \\ \hline 3 & 0.027 \\ \hline 4 & 0.004 \\ \hline 5 & 0.001 \end{array} $$ (a) Verify that this is a discrete probability distribution. (b) Draw a graph of the probability distribution. Describe the shape of the distribution. (c) Compute and interpret the mean of the random variable \(X\). (d) Compute the standard deviation of the random variable \(X\). (e) What is the probability that a randomly selected individual 15 years or older was involved in two marriages? (f) What is the probability that a randomly selected individual 15 years or older was involved in at least two marriages?

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.