Problem 16 - Algebra 2

Question

For Exercises 15-17, use the following information. A binomial distribution has a 60\(\%\) rate of success. There are 18 trials. What is the probability that there will be 12 failures?

Step-by-Step Solution

Answer

The probability of 12 failures is approximately 0.0629.

Step 1: Identify Parameters of the Binomial Distribution

In a binomial distribution, we need to identify the number of trials \(n\), the probability of success \(p\), and the probability of failure \(q\). In this problem, there are 18 trials, so \(n = 18\). The probability of success is 60\%, so \(p = 0.6\). Since the probability of failure is 1 minus the probability of success, we have \(q = 1 - p = 0.4\).

Step 2: Determine What is Asked

The problem asks for the probability of exactly 12 failures. To find this, we first determine the number of successes needed. With 18 trials in total, if there are 12 failures, then there are 18 - 12 = 6 successes.

Step 3: Use the Binomial Probability Formula

The formula for the probability of exactly \(k\) successes in \(n\) binomial trials is given by: \[P(X = k) = \binom{n}{k} \, p^k \, q^{n-k}\] Here, \(\binom{n}{k}\) represents the binomial coefficient, \(p\) is the probability of success, and \(q\) is the probability of failure.

Step 4: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{n}{k}\) for \(n = 18\) and \(k = 6\) is calculated by: \[\binom{18}{6} = \frac{18!}{6!(18-6)!} = \frac{18!}{6! \, 12!}\] Calculating each factorial and simplifying, we find that \(\binom{18}{6} = 18564\).

Step 5: Calculate the Probability

Now compute the probability using the formula: \[P(X = 6) = \binom{18}{6} \, (0.6)^6 \, (0.4)^{12}\]Substitute the known values: - \(\binom{18}{6} = 18564\)- \(p = 0.6\), so \(p^6 = (0.6)^6\)- \(q = 0.4\), so \(q^{12} = (0.4)^{12}\)After computation, the probability is approximately 0.0629.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena.
In simplistic terms, it quantifies how likely events are to occur. This can help us make predictions about random outcomes.
In probability theory, randomness is handled using a probability space, which includes:

• The sample space (all possible outcomes of an experiment)
• A set of events (specific outcomes from the sample space)
• Probabilities associated with each event

Understanding probability theory is crucial for interpreting and calculating likely outcomes, such as in the case of a binomial distribution.

Binomial Probability Formula

The binomial probability formula is fundamental for calculating the likelihood of a specific number of successes in a series of independent trials.
This formula applies to scenarios where there are only two possible outcomes in each trial, often referred to as "success" and "failure."
The formula is:\[P(X = k) = \binom{n}{k} \, p^k \, q^{n-k}\]where:

• \(n\) is the total number of trials
• \(k\) is the number of successful outcomes we want to find the probability for
• \(p\) is the probability of success on an individual trial
• \(q = 1 - p\) is the probability of failure
• \(\binom{n}{k}\) is the binomial coefficient

This formula is widely used in situations with fixed numbers of trials, helping predict probabilities of varied outcomes.

Binomial Coefficient

The binomial coefficient, denoted as \(\binom{n}{k}\), is a mathematical expression used to determine the number of different ways \(k\) successes can occur in \(n\) trials.
This is a critical component of the binomial probability formula.
To calculate it, use the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This expression uses factorials, which are products of all positive integers up to a given number.
For instance, \(!6 = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
The binomial coefficient allows us to understand all possible arrangements of successes within the trials, crucial for accurately computing probabilities.

Probability of Success and Failure

In the context of a binomial distribution, each trial has two possible outcomes: success and failure. Understanding their probabilities is essential for working with the binomial probability formula.
- The **probability of success** (often represented as \(p\)) is the likelihood that a single trial will result in a success.
It's a value between 0 and 1 and must be consistent across all trials.
- The **probability of failure** (designated as \(q\)) is calculated as \(q = 1 - p\).
This is because success and failure are mutually exclusive outcomes.Having both these probabilities enables us to predict outcomes over multiple trials, foundational for the calculations involved in binomial distribution problems.