Textbook Question
"Problems 17–28: In each problem, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in n independent trials of the experiment.
n = 8, p = 0.35, x = 3"
3
views
5. Binomial Distribution & Discrete Random Variables
Binomial Distribution
1:59 minutesProblem 6.2.22
Textbook Question
"Problems 17–28: In each problem, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in n independent trials of the experiment.
n = 20, p = 0.6, x = 17"
Verified step by step guidance1
Identify the parameters of the binomial distribution: number of trials \(n = 20\), probability of success in each trial \(p = 0.6\), and the number of successes \(x = 17\).
Recall the binomial probability formula:
\[P(X = x) = \binom{n}{x} p^{x} (1-p)^{n-x}\]
where \(\binom{n}{x}\) is the binomial coefficient representing the number of ways to choose \(x\) successes from \(n\) trials.
Calculate the binomial coefficient \(\binom{20}{17}\) using the formula:
\[\binom{n}{x} = \frac{n!}{x! (n-x)!}\]
which counts the number of combinations of 17 successes in 20 trials.
Compute the powers of the probabilities:
\[p^{x} = (0.6)^{17}\]
and
\[(1-p)^{n-x} = (0.4)^{3}\]
since \$1-p$ is the probability of failure.
Multiply the binomial coefficient by these powers to find the probability:
\[P(X=17) = \binom{20}{17} \times (0.6)^{17} \times (0.4)^{3}\]
This gives the probability of exactly 17 successes in 20 trials.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Probability Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is used when outcomes are binary (success/failure), and the probability of success remains constant across trials.
Recommended video:
Guided course
06:39
Calculating Probabilities in a Binomial Distribution
Parameters of the Binomial Distribution (n, p, x)
In a binomial experiment, n represents the total number of trials, p is the probability of success on each trial, and x is the number of successes for which we want to find the probability. These parameters define the shape and probabilities of the distribution.
Recommended video:
Guided course
03:28Mean & Standard Deviation of Binomial Distribution
Binomial Probability Formula
The probability of exactly x successes in n trials is calculated using the formula: P(X = x) = C(n, x) * p^x * (1-p)^(n-x), where C(n, x) is the number of combinations of n items taken x at a time. This formula combines combinatorics and probability powers.
Recommended video:
Guided course
06:39Calculating Probabilities in a Binomial Distribution