Problem 11 - Algebra 2

Question

Find the probability of \(x\) successes in \(n\) trials for the given probability of success \(p\) on each trial. $$ x=5, n=10, p=0.1 $$

Step-by-Step Solution

Answer

The probability of achieving exactly 5 successes in 10 trials, with the success probability of 0.1 on each trial, is approximately 0.0013

Step 1: Writing the formula

To solve this problem, the Binomial Probability Formula should be used: \( P(x;n,p) = C(n, x) * p^x * (1-p)^{n-x} \), where \( P(x;n,p) \) stands for the probability of achieving exactly \( x \) successes in \( n \) trials, \( p \) stands for the probability of success on each trial, and \( C(n,x) \) stands for the binomial coefficient.

Step 2: Substituting the given values

Substitute the given values into the formula: \( P(5;10,0.1) = C(10, 5) * (0.1)^5 * (1-0.1)^{10-5} \)

Step 3: Calculating the binomial coefficient

Calculate the binomial coefficient: \( C(10, 5) = \frac{10!}{5!(10-5)!} = 252 \)

Step 4: Putting it all together

Insert the calculated binomial coefficient into the formula and calculate the result: \( P(5;10,0.1) = 252 * (0.1)^5 * (0.9)^5 = 0.001319938 \)

Key Concepts

Understanding Binomial Coefficient

The binomial coefficient is a crucial part of the binomial probability formula. It helps us determine the number of ways we can choose certain outcomes from a set of trials. For example, in the given problem of finding 5 successes in 10 trials, the binomial coefficient represents how many different combinations of 5 successes can occur out of 10 trials.

Mathematically, it is denoted as \( C(n, x) \) or sometimes as \( \binom{n}{x} \). It is calculated using the formula: - \[ C(n, x) = \frac{n!}{x!(n-x)!} \] where \( n! \) ("n factorial") is the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

• In the problem, \( n = 10 \) and \( x = 5 \), so \( C(10, 5) \) is computed by: \( \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = 252 \).

This tells us there are 252 ways to achieve 5 successes in a series of 10 trials.

Probability of Success

Probability of success is the likelihood that any single trial will result in the desired outcome. It is an essential component of binomial distribution because it determines the overall chance of achieving a specific number of successes across multiple trials.

In the example provided, the probability of each trial resulting in success is given as \( p = 0.1 \). This means there is a 10% chance of success every time a trial is performed.

• For failures, the probability is \( 1-p \), which calculates to 90% in this scenario.

The binomial formula uses this information to calculate the overall probability by raising \( p \) to the power of \( x \) (the number of successes) and \( (1-p) \) to the power of \( n-x \) (the number of failures).This method combines individual probabilities to understand the likelihood of specific outcomes.

Exploring Binomial Distribution

Binomial distribution is a statistical method that describes the probability of obtaining a fixed number of successes in a set number of trials. All trials have the same probability of success, making it a common distribution used in statistics for likelihood estimation.

This distribution is applicable in scenarios involving "yes/no" or "success/failure" outcomes.

• Each trial is independent, meaning the outcome of one trial doesn't affect another.
• Used when there are two mutually exclusive outcomes.

The formula to find binomial probability is a combination of the binomial coefficient, probability of success, and probability of failure:- \[ P(x;n,p) = C(n, x) * p^x * (1-p)^{n-x} \]In our problem, substituting the numbers helps find that there is a very low (around 0.13%) chance of exactly 5 successes out of 10 when success has a small probability of 0.1 in each trial. This distribution provides insight into the likelihood of obtaining various outcomes of interest over repeated tests, offering valuable information for decision-making and analysis.