Problem 38 - Algebra 2

Question

CHALLENGE Find the probability of exactly \(m\) successes in \(n\) trials of a binomial experiment where the probability of success in a given trial is \(p .\)

Step-by-Step Solution

Answer

The probability of exactly \( m \) successes in \( n \) trials is given by \( P(X = m) = \binom{n}{m} p^m (1-p)^{n-m} \).

Step 1: Identify the Problem Type

This problem is a typical application of the binomial probability formula. We are trying to find the probability of getting exactly \( m \) successes in \( n \) trials.

Step 2: Use the Binomial Probability Formula

The formula to find the probability of exactly \( m \) successes in \( n \) trials is given by:\[P(X = m) = \binom{n}{m} p^m (1-p)^{n-m}\] where \( \binom{n}{m} \) is the binomial coefficient, which can be calculated by \( \frac{n!}{m!(n-m)!} \).

Step 3: Calculate the Binomial Coefficient

Calculate the binomial coefficient \( \binom{n}{m} \) using the formula:\[\binom{n}{m} = \frac{n!}{m!(n-m)!}\] This gives the number of ways to choose \( m \) successes out of \( n \) trials.

Step 4: Calculate the Success Probability Component

Compute \( p^m \), which is the probability of having success in exactly \( m \) trials. This accounts for successful outcomes.

Step 5: Calculate the Failure Probability Component

Compute \((1-p)^{n-m}\), which is the probability of having a failure in the remaining \( n-m \) trials.

Step 6: Combine the Components

Multiply the results to form the complete probability for exactly \( m \) successes:\[P(X = m) = \binom{n}{m} p^m (1-p)^{n-m}\]

Key Concepts

Binomial Coefficient

In the context of binomial probability, the binomial coefficient plays a crucial role. It helps us determine the number of ways we can achieve exactly \( m \) successes in \( n \) trials. This is vital because, in a binomial experiment, the order of successes and failures matters, and the binomial coefficient lets us compute the possible arrangements efficiently.

The binomial coefficient is calculated using the formula:

• \( \binom{n}{m} = \frac{n!}{m!(n-m)!} \)

The exclamation mark in the formula denotes a factorial, which we'll discuss later on. Essentially, \( \binom{n}{m} \) gives us a specific count of ways to choose \( m \) successes from \( n \) possible trials, without caring about the order of selection. For example, if we had 5 trials and wanted exactly 3 to be successful, \( \binom{5}{3} \) would give us the number of combinations possible.

Probability of Success

The probability of success, denoted by \( p \), is another cornerstone of binomial probability problems. When we perform a binomial experiment, each trial has two possible outcomes: success or failure.

Here's what you need to know about the probability of success:

• It's the likelihood of a favorable outcome occurring in a single trial.
• If the probability of success is \( p \), then the probability of failure in any given trial is \( 1-p \).

In the binomial probability formula, the probability of having exactly \( m \) successes across \( n \) trials is represented by \( p^m \). This term reflects the likelihood of success occurring exactly \( m \) times during the entire experiment. Thus, understanding and calculating \( p \) is essential for solving binomial problems.

Number of Trials

When dealing with binomial probability, one of the first steps is identifying the number of trials, denoted as \( n \). Each trial in a binomial experiment represents an individual occurrence of the process being studied.

Here's why the number of trials is important:

• It sets the framework for how many times the event will be repeated.
• The number \( n \) influences both the number of possible outcomes and the structure of the binomial formula.

For example, if you are flipping a coin 10 times, each flip is a trial, so \( n = 10 \). Understanding the total number of trials is crucial as it directly impacts the computation of the binomial coefficient and the overall probability.

Factorial

Factorials are a fundamental concept used in calculating binomial coefficients. In mathematics, a factorial, represented by an exclamation mark (\(!\)), is the product of all positive integers up to a specified number, \( n \).

Consider these key aspects of factorials:

• For a positive integer \( n \), the factorial is given by \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \).
• Factorials are utilized to calculate permutations and combinations, critical in binomial probability calculations.

In our problem, the factorial is used in the computation of the binomial coefficient:

• \( \binom{n}{m} = \frac{n!}{m!(n-m)!} \)

Here, factorials help us determine the number of ways we can distribute successes and failures across trials. Knowing how to calculate factorials is vital to understanding and applying the binomial probability formula.