Problem 6 - Algebra and Trigonometry

Question

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. All successes

Step-by-Step Solution

Answer

The probability of all successes is 0.16807.

Step 1: Understand the Binomial Probability Formula

In a binomial experiment, the probability of getting exactly \( k \) successes in \( n \) independent trials, each with a probability of success \( p \), is given by the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]In this problem, \( n = 5 \) trials, \( p = 0.7 \), and \( q = 0.3 \), and we need to find the probability of all successes, i.e., \( k = 5 \).

Step 2: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{5}{5} \) represents the number of ways to choose 5 successes from 5 trials and is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Substitute \( n = 5 \) and \( k = 5 \):\[\binom{5}{5} = \frac{5!}{5! \, 0!} = 1\]

Step 3: Compute the Probability Term for All Successes

Substitute \( k = 5 \), \( p = 0.7 \), and \( q = 0.3 \) into the probability term \( p^k (1-p)^{n-k} \):\[p^k = (0.7)^5 = 0.16807\]\[(1-p)^{n-k} = (0.3)^0 = 1\]

Step 4: Calculate the Probability for All Successes

Now that we have \( \binom{5}{5} = 1 \), \( (0.7)^5 = 0.16807 \), and \( (0.3)^0 = 1 \), multiply these together to find the probability of all successes:\[P(X = 5) = \binom{5}{5} (0.7)^5 (0.3)^0 = 1 \times 0.16807 \times 1 = 0.16807\]

Key Concepts

Binomial Experiment

A binomial experiment is a statistical experiment where there are exactly two possible outcomes in each trial: success or failure. This type of experiment is characterized by a fixed number of trials, denoted as \( n \). Each trial is independent, meaning the outcome of one trial does not affect another. Additionally, the probability of success, denoted as \( p \), remains constant for each trial. Such experiments are commonly used to model situations like flipping a coin multiple times or checking pass rates in a batch of products.

For example, the problem given involves five independent trials, where the probability of success in each trial is \( 0.7 \). Since there are only two outcomes in each trial (success or failure), this satisfies the conditions of a binomial experiment. These trials aim to find the probability of having all successes, which is an application of the binomial probability distribution.

The understanding of binomial experiments is crucial for solving problems involving repeated independent trials, as it helps in determining the likelihood of achieving a set number of successes in those trials.

Probability of Success

The probability of success, often represented by \( p \), is a critical component of a binomial experiment. It measures the likelihood of a single trial resulting in a success and is a constant throughout all the trials of the experiment. In the context of our example, where \( p = 0.7 \), this means that each trial has a 70% chance of being successful.

Alongside the probability of success, we also have the probability of failure, represented by \( q \), where \( q = 1 - p \). In our scenario, \( q = 0.3 \), indicating a 30% chance of failure in each trial.

The probability of success is crucial in determining the overall probability of certain outcomes in multiple trials. For example, to determine the probability of achieving \( k \) successes in \( n \) trials, we use it along with other factors in the binomial probability formula. This calculation gives us an insight into how likely a certain number of successes is, helping make informed decisions based on these probabilities.

Binomial Coefficient

The binomial coefficient is a mathematical concept that counts the number of ways \( k \) successes can occur in \( n \) independent trials. It is denoted by \( \binom{n}{k} \) and computed using the formula:

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

Here, \( n! \) is the factorial of \( n \), and it represents the product of all positive integers less than or equal to \( n \).

In the provided problem, we are exploring the scenario of achieving all successes in five trials (\( k = 5 \) successes in \( n = 5 \) trials). Hence, the binomial coefficient \( \binom{5}{5} \) is calculated to be 1, indicating only one possible way to achieve 5 successes in 5 trials.

Understanding the binomial coefficient is central to working with binomial probabilities, as it helps in determining the weight of each outcome in the distribution. It shows the various ways successes can be arranged across trials, thereby structuring the foundation for calculating the probability of observing a specified number of successes in a given set of independent trials.