Problem 9 - Algebra 2

Question

Find the probability of \(x\) successes in \(n\) trials for the given probability of success \(p\) on each trial. $$ x=4, n=8, p=0.3 $$

Step-by-Step Solution

Answer

To find the solution, use the binomial probability formula plug in the given numbers \(n=8, x=4, p=0.3\), simplify the formula and calculate the probability.

Step 1: Plug into the binomial formula

First, we'll plug the given values into the binomial probability formula: \(x=4, n=8, p=0.3\). This gives us \( P(4;8,0.3) = \binom{8}{4} \cdot 0.3^{4} \cdot (1-0.3)^{8-4}\).

Step 2: Calculate 'n choose x'

\'8 choose 4\ is calculated as \( \frac{8!}{4!(8-4)!}\), which simplifies to 70. Substitute this value back into the formula.

Step 3: Simplify

Now simplify further: \(P(4;8,0.3)= 70 \cdot 0.3^{4} \cdot 0.7^{4}\) .

Step 4: Calculate the probability

Perform the calculations to find the probability.

Key Concepts

Probability of Success

In the context of a binomial distribution, the probability of success refers to the likelihood that a single trial results in the desired outcome. In many instances, this is represented by the variable \(p\).
For example, when rolling a fair six-sided die, if we define a success as rolling a 4, the probability \(p\) of success is \(\frac{1}{6}\).
However, in our specific problem, the probability of success is given as \(0.3\), implying that there is a 30% chance of the desired result occurring in each trial. This probability remains constant across all trials.

• It is important to understand that a success does not always mean a positive outcome; it merely represents one that meets the specific criteria being measured.
• A consistent probability \(p\) over numerous trials is what makes binomial analysis possible.

Binomial Formula

The binomial formula is a powerful tool used to determine the probability of achieving exactly \(x\) successes in \(n\) trials. This formula is expressed as:\[ P(x; n, p) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x} \]Where:

• \(P(x; n, p)\) is the probability of \(x\) successes.
• \(\binom{n}{x}\) represents the number of combinations, or ways, \(x\) successes can occur in \(n\) trials.
• \(p^x\) corresponds to the probability of obtaining success \(x\) times.
• \((1-p)^{n-x}\) indicates the probability of \(n-x\) failures.

In applying this formula to real-world scenarios, it provides a structured approach to predict outcomes.
Using the formula, a student can confidently calculate the likelihood of a certain number of successes, provided they know the values of \(n\), \(x\), and \(p\).
This formula remains a cornerstone of probability theory due to its robustness and simplicity.

Combinatorial Analysis

Combinatorial analysis deals with counting, arranging, and listing elements within sets, and is essential to understanding probabilities in binomial distributions. Often represented by the binomial coefficient, it calculates the number of ways to choose \(x\) successes out of \(n\) trials.
The binomial coefficient is denoted as \(\binom{n}{x}\) and can be calculated using the formula:\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]

• Here, \(n!\) (n factorial) represents the product of all positive integers up to \(n\).
• \(x!\) is the factorial of \(x\), and \((n-x)!\) is the factorial of \(n-x\).

In simpler terms, the binomial coefficient quantifies how many different ways you can arrange a given number of successes in a sequence of trials.
This combinatorial insight is critical as it allows us to understand all possible ways that events can unfold, making it a foundational element in both probability and statistics. By mastering this concept, students gain a deeper insight into the dynamic nature of probability analysis.