Problem 3 - 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. No successes

Step-by-Step Solution

Answer

The probability of no successes is approximately 0.00243.

Step 1: Identify the Parameters

The problem provides us with a binomial experiment with the probability of success denoted as \( p = 0.7 \), and thus, the probability of failure is \( q = 1 - p = 0.3 \). The number of trials is \( n = 5 \).

Step 2: Apply the Binomial Probability Formula

The probability of getting exactly \( k \) successes in \( n \) trials of a binomial experiment is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where \( \binom{n}{k} \) is the binomial coefficient, \( p \) is the probability of success, and \( (1-p) \) is the probability of failure.

Step 3: Calculate the Binomial Coefficient

For \( k = 0 \) successes, the binomial coefficient is:\[ \binom{5}{0} = \frac{5!}{0!(5-0)!} = 1 \]

Step 4: Calculate the Probability Component

Use the probability of success \( p = 0.7 \) and the probability of failure \( q = 0.3 \):\[ p^k = (0.7)^0 = 1 \]\[ (1-p)^{n-k} = (0.3)^5 \]

Step 5: Compute the Probability

Now substitute the values into the binomial probability formula for \( k = 0 \):\[ P(X = 0) = \binom{5}{0} (0.7)^0 (0.3)^5 = 1 \times 1 \times 0.3^5 \]\[ P(X = 0) = 0.3^5 = 0.00243 \]

Step 6: Conclusion

The probability of having no successes in five trials of this binomial experiment is \( 0.00243 \).

Key Concepts

Binomial Experiment

A binomial experiment is a type of probability experiment that consists of a fixed number of trials. Each trial has exactly two possible outcomes: success or failure. In a binomial experiment, the probability of success and failure remains constant throughout the trial. Here are some key characteristics of a binomial experiment:

• The number of trials \( n \) is fixed and known in advance.
• Each trial is independent, meaning the outcome of one trial does not affect others.
• There are only two possible outcomes for each trial: "success" with probability \( p \), and "failure" with probability \( q = 1 - p \).

In our exercise, we have a binomial experiment with \( n = 5 \), which means five trials are conducted independently. The probability of success \( p \) and failure \( q \) remains constant as 0.7 and 0.3, respectively.

Probability of Success

The probability of success in a binomial experiment is denoted by \( p \). This is a fixed probability assigned to achieving a successful outcome in one individual trial. The exercise specifies that the probability of success \( p \) is 0.7. Here's what this means:

• A success is any outcome that you are counting as a favorable result in the context of your problem. This can vary depending on the situation.
• In each of the five trials, there is a 70% chance that the event will result in success.

The complementary probability, which is the probability of failure \( q \), is given by \( q = 1 - p = 0.3 \), meaning there is a 30% chance of not achieving a successful outcome. Understanding the probability of success is crucial for calculating the probability of different outcomes in a binomial experiment.

Binomial Coefficient

The binomial coefficient, represented as \( \binom{n}{k} \), is a key component when calculating probabilities in a binomial experiment. It determines how many different ways \( k \) successes can occur in \( n \) trials. The binomial coefficient is calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

• \( n! \) denotes the factorial of the number of trials, \( n \).
• \( k! \) denotes the factorial of the number of successes you are calculating for, \( k \). For no successes, \( k = 0 \).

In the provided exercise, \( \binom{5}{0} = 1 \), showing there's exactly one way to achieve zero successes across all trials. This step is crucial because it gives the number of possible combinations for the given number of successes, which is then used in the binomial probability formula to find event probabilities.