Problem 8 - Algebra 2

Question

Find the probability of \(x\) successes in \(n\) trials for the given probability of success \(p\) on each trial. $$ x=3, n=8, p=0.3 $$

Step-by-Step Solution

Answer

The probability of achieving exactly three successes in eight trials, with the probability of success on each trial being 0.3, is calculated using the binomial probability formula. The detailed steps follow the identification of the values, application of the binomial probability formula, calculation of the binomial coefficient, and the plug-in of the values into the computed formula.

Step 1: Identify the values in the problem

Here, the number of successes x is 3, the number of trials n is 8, and the probability of success on each trial p is 0.3.

Step 2: Apply the binomial probability formula

Using the formula for binomial probability, \(P(x;n,p) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}\), we can substitute the values into the formula which will look like this: \(P(3;8,0.3) = \binom{8}{3} \cdot (0.3)^3 \cdot (1-0.3)^{8-3}\).

Step 3: Calculate the binomial coefficient

The binomial coefficient, \(\binom{8}{3} = \frac{8!}{3!(8-3)!} = 56\), where '!' denotes a factorial. Now substitute it back into our formula.

Step 4: Plug the values into the computed formula

So now we have \(P(3;8,0.3) = 56 \cdot (0.3)^3 \cdot (0.7)^5\). Calculating the values, we arrive at the probability P.

Key Concepts

Understanding Probability Theory

Probability theory is a branch of mathematics that deals with predicting the likelihood of certain events happening. One part of this field is concerned with determining how likely it is for a specific outcome to occur out of all possible outcomes. In simpler terms, it's like asking, "What are the odds of this happen?"

In this context, we are specifically interested in **binomial probability**, which refers to the chance of obtaining a fixed number of successes in a series of independent and identical trials, each with the same probability of success. In our example, this means calculating the probability of achieving exactly 3 successes in 8 attempts, with each attempt having a 0.3 chance of success.

Unveiling the Binomial Coefficient

A crucial part of calculating binomial probabilities is understanding the binomial coefficient. This coefficient tells us how many different ways we can achieve the desired number of successes in a given number of trials. It is represented as \( \binom{n}{x} \), pronounced as "n choose x."

The formula for calculating the binomial coefficient is \( \binom{n}{x} = \frac{n!}{x!(n-x)!} \). Here, the exclamation point represents a **factorial**, which we will explain more in the following section. In our problem, to find \( \binom{8}{3} \), we substitute the values into the formula, getting \( \frac{8!}{3!(8-3)!} = 56 \). What this signifies is that there are 56 different sequences to achieve 3 successes in 8 attempts.

Exploring Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a certain number of successes will take place in a certain number of trials, with each trial having the same probability of success. It is key in situations where only two outcomes exist for each trial, such as success or failure, yes or no.

The core characteristic of the binomial distribution is governed using the formula:

P(x;n,p) = \( \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x} \)

In this formula, each component contributes to the calculation:

• \( \binom{n}{x} \): represents the number of combinations for choosing x successes in n trials.
• \( p^x \): gives the probability of having successes through x trials.
• \( (1-p)^{n-x} \): represents the probability of the remaining trials resulting in failures.

This formula provides a complete picture of the probability of achieving a desired number of successes.

Deciphering Factorials

The concept of a factorial is vital when dealing with probabilities and permutations. When you see the notation !, it refers to a **factorial**. A factorial of a non-negative integer is the product of all positive integers less than or equal to that number. For instance, 5! means 5 × 4 × 3 × 2 × 1, which equals 120.

Factorials are pivotal in calculating the binomial coefficient, helping us determine combinations of success. In our example, we calculated \( 8! \) and \( 3! \) to find the binomial coefficient \( \binom{8}{3} \). Understanding how to compute factorials is crucial to mastering binomial probability and ensuring accuracy in calculating probabilities based on combinations.