/*! 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 109 Calculate the requested binomial... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 109

Calculate the requested binomial probability. Find \(P(X=2)\) if \(X\) is a binomial random variable with \(n=6\) and \(p=0.3\).

Short Answer

Expert verified
The binomial probability \(P(X=2)\) is approximately 0.3241.

Step by step solution

01

Identify known values

First, identify and write down the known values from the exercise. Here, \(n = 6\), \(k = 2\), and \(p = 0.3\). You can calculate \(q\) from \(p\), as \(q = 1 - p\). So, \(q = 1 - 0.3 = 0.7\).
02

Calculate the binomial coefficient

The binomial coefficient \(C(n, k)\) is calculated as \(n!/k!(n-k)!\). So, \(C(6, 2) = 6!/(2!(6-2)!) = 15\).
03

Substitute known values into the formula and calculate

Substitute known values into the binomial formula. This gives \(P(X=2) = 15 \cdot (0.3^2) \cdot (0.7^4) \approx 0.3241\).

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 Random Variable
Understanding the concept of a binomial random variable is vital to grasping binomial probability. A binomial random variable represents the number of successes in a fixed number of independent trials in a binomial experiment. Each trial can only result in success (with probability p) or failure (with probability q=1-p), and the probability of success remains the same for each trial.

An example of a binomial random variable is the number of heads when flipping a coin multiple times. When calculating P(X=2) where X is a binomial random variable with n=6 trials and success probability p=0.3, we are essentially looking to find the probability of exactly 2 successes (like getting heads in our example) in 6 trials (flips).
Binomial Coefficient
The binomial coefficient, often denoted as C(n, k) or \( \binom{n}{k} \), describes the number of ways to choose k successes from n trials. It's a central concept in combinatorics and appears in the formula to calculate binomial probabilities.

The coefficient is calculated using factorial notation where n! denotes the product of all positive integers up to n: C(n, k) = \( \frac{n!}{k!(n-k)!} \). Hence, for our example, C(6, 2) helps us to determine in how many different ways we can achieve 2 successes in 6 trials, which is crucial before we further apply the binomial formula.
Binomial Formula
The binomial formula, also known as the binomial probability formula, gives us the probability of getting exactly k successes in n independent Bernoulli trials. It is expressed as:
P(X = k) = C(n, k) * p^k * q^(n-k),
where X is the binomial random variable, p is the probability of success on each trial, and q is the probability of failure.

In the example given, after calculating the binomial coefficient C(6, 2) as 15, we can apply the binomial formula by substituting the values for p, q, and k to find the probability P(X=2). Plugging in the values gives us 15 * (0.3^2) * (0.7^4), which simplifies to approximately 0.3241. This formula is a powerful tool for calculating probabilities in scenarios that fit the criteria of a binomial experiment.

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

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.4, P(B)=0.3\), and \(P(A\) and \(B)=0.1\). Find \(P(\operatorname{not} A)\)

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B},\) we have \(P(A)=0.8, P(B)=0.4\) and \(P(A\) and \(B)=0.25\). Are events A and B disioint?

The word "free" is contained in \(4.75 \%\) of all messages, and \(3.57 \%\) of all messages both contain the word "free" and are marked as spam. (a) What is the probability that a message contains the word "free", given that it is spam? (b) What is the probability that a message is spam, given that it contains the word "free"?

In a certain board game participants roll a standard six-sided die and need to hit a particular value to get to the finish line exactly. For example, if Carol is three spots from the finish, only a roll of 3 will let her win; anything else and she must wait another turn to roll again. The chance of getting the number she wants on any roll is \(p=1 / 6\) and the rolls are independent of each other. We let a random variable \(X\) count the number of turns until a player gets the number needed to win. The possible values of \(X\) are \(1,2,3, \ldots\) and the probability function for any particular count is given by the formula $$P(X=k)=p(1-p)^{k-1}$$ (a) Find the probability a player finishes on the third turn. (b) Find the probability a player takes more than three turns to finish.

Probability of being in each cell of a two-way table $$\begin{array}{c|cc} \hline & A & \text { not } A \\ \hline B & 0.2 & 0.4 \\ \operatorname{not} B & 0.1 & 0.3 \end{array}$$ State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.)Roll two (six-sided) dice. Let \(\mathrm{A}\) be the event that the first die is a 3 and \(\mathrm{B}\) be the event that the sum of the two dice is 8 .
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

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.