Problem 12 - 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. At most three failures

Step-by-Step Solution

Answer

The probability of at most three failures is approximately 0.969.

Step 1: Define the Problem

We need to find the probability of at most three failures in five independent binomial trials, where the probability of failure for each trial is \( q = 0.3 \).

Step 2: Binomial Probability Formula

The probability of obtaining exactly \( k \) successes in \( n \) trials of a binomial experiment is given by the formula: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient calculated as \( \frac{n!}{k!(n-k)!} \).

Step 3: Calculate Probabilities

Calculate the probabilities for 0, 1, 2, and 3 failures (equivalent to 5, 4, 3, and 2 successes respectively):- \( P(X = 5) = \binom{5}{0} (0.7)^5 (0.3)^0 = 0.16807 \)- \( P(X = 4) = \binom{5}{1} (0.7)^4 (0.3)^1 = 0.36015 \)- \( P(X = 3) = \binom{5}{2} (0.7)^3 (0.3)^2 = 0.3087 \)- \( P(X = 2) = \binom{5}{3} (0.7)^2 (0.3)^3 = 0.1323 \)

Step 4: Sum the Probabilities

Sum the probabilities for 0, 1, 2, and 3 failures:\[ P(X \leq 3) = P(X=5) + P(X=4) + P(X=3) + P(X=2) \]\[ P(X \leq 3) = 0.16807 + 0.36015 + 0.3087 + 0.1323 = 0.96922 \]

Step 5: Conclusion

The probability of at most three failures in five trials with \( p = 0.3 \) (failure probability) is \( 0.96922 \).

Key Concepts

Binomial Coefficient

In a binomial experiment, the binomial coefficient is a key component that appears in the probability formula. It is represented by \( \binom{n}{k} \) and is used to determine the number of ways \( k \) successes can occur in \( n \) trials. The binomial coefficient can be calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).
• \( k! \) is the factorial of \( k \).
• The product \((n-k)!\) stands for the factorial of \((n-k)\).

The calculation of these coefficients helps in figuring out how many combinations there are without considering the arrangement order. This is crucial in solving problems involving binomial probability distributions.

Independent Trials

Understanding independent trials is fundamental to performing binomial experiments. In such experiments, each trial is independent of the others. This means the outcome of one trial does not affect the outcome of another.
Here are some important aspects of independent trials:

• The probability of success or failure remains constant throughout the trials.
• Each trial can result in either a success or a failure, with probabilities defined as \( p \) and \( q \), respectively.

This independence allows us to apply the binomial probability formula correctly. For example, if we have five trials, each trial outcome has no bearing on the other trials' results, ensuring that the probability calculations are valid.

Probability of Failure

The probability of failure in a binomial experiment is denoted by \( q \). It represents the likelihood that a single trial results in a failure. In most problems, you will find \( q \) by subtracting the probability of success, \( p \), from 1:
\[q = 1 - p\]For instance, in the given exercise, the probability of success \( p \) is 0.7, making the probability of failure \( q = 1 - 0.7 = 0.3 \). It is important to know this value because it is used in the binomial probability formula to calculate different outcomes:

• It appears in the term \( q^{(n-k)} \) in the formula \( P(X = k) = \binom{n}{k} p^k q^{n-k} \).
• Understanding \( q \) helps determine the likelihood of occurrences where failures are involved, like calculating at most three failures in an experiment.

Probability of Success

The probability of success, \( p \), signifies the chance that a single trial results in success. In a binomial experiment, this probability stays constant through all trials.
Consider these key points:

• Every trial in the experiment has the probability \( p \) of being successful.
• In the exercise you are looking at, \( p \) is given as 0.7.

This value is critical in the binomial probability formula, contributing to the term \( p^k \), where \( k \) is the number of successes. Accurately understanding and using \( p \) is imperative to solving problems in the context of independent trials, ensuring the accuracy of calculations regarding expected outcomes.