Problem 68 - Algebra 2

Question

For the given probability of success \(p\) on each trial, find the probability of \(x\) successes in \(n\) trials. $$ x=7, n=8, p=0.7 $$

Step-by-Step Solution

Answer

Calculate the final solution obtained in Step 3 to find the probability of getting 7 successes in 8 trials when the probability of success on each trial is 0.7.

Step 1: Identify Known Values

From the problem, we know the following:- Number of trials (\(n\)) = 8- Probability of success (\(p\)) = 0.7- Number of successes (\(x\)) = 7

Step 2: Calculate Combinations

Use the combinations formula \(C(n,k) = n! / [(n - k)! k!]\) where '!' means factorial, so we have \(C(n,k) = 8! / [(8 - 7)! 7!]\)

Step 3: Compute Binomial Probability

Insert the values into the binomial probability formula: \[ P(X=k)=C(n,k)*(p^k)*(1-p)^{n-k}\]\[ P(X=7)=C(8,7)*(0.7^7)*(1-0.7)^{8-7} \]

Key Concepts

Probability of Success

In probability theory, the concept of "probability of success" is key to understanding binomial experiments. When we talk about a probability of success, denoted as \( p \), we refer to the likelihood that one trial in our set of trials will result in a desired outcome. For example, if we are flipping a coin and are interested in landing a "head," the probability of success is 0.5 for a fair coin.

In the given exercise, the probability of success \( p \) is 0.7, meaning there's a 70% chance for the trial's outcome to be successful. It's important to understand that this probability remains the same for each trial in the experiment. This constancy is what makes the experiment binomial. Each trial is independent of the others, and the probability of success on each trial is constant.

• Each trial is independent
• The probability of success \( p \) does not change

Combinations Formula

To solve problems using binomial probability, we often need to use the combinations formula. This formula is crucial because it calculates how many ways we can choose \( x \) successes out of \( n \) trials. The combinations formula is given by \( C(n, k) = \frac{n!}{(n-k)!k!} \). Here, \( n \) is the total number of trials, \( k \) is the number of successful trials we're interested in, and \( ! \) denotes factorial, which we'll cover next.

In our example, we are looking for 7 successes in 8 trials. We use the formula as follows, \( C(8, 7) = \frac{8!}{(8-7)!7!} \). This will help us find the total number of ways to achieve exactly 7 effective outcomes when conducting 8 trials.

• The combinations formula calculates ways to choose successful outcomes
• It requires knowledge of factorials, which we'll discuss shortly

Factorial

Factorials are a fundamental concept when working with combinations and binomial probabilities. Mathematically, a factorial is the product of an integer and all the positive integers below it. It is denoted by an exclamation mark next to the number. For example, \( 4! \) means \( 4 \times 3 \times 2 \times 1 \), which equals 24.

Factorials help simplify calculations when determining the number of combinations in binomial problems. In our exercise, to compute \( C(8, 7) \), you need to evaluate the factorials in the formula \( \frac{8!}{(8-7)!7!} \). Calculating this will help identify how many ways 7 successes can occur within 8 trials. Factorials grow very quickly, so it's important to use them correctly when calculating larger numbers.

• A factorial is a product of all integers from 1 to that number
• Understanding factorials is key to using the combinations formula effectively