Problem 14 - 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 0.96922.

Step 1: Identify the Parameters

We have a binomial distribution with parameters: number of trials \(n = 5\), probability of success \(p = 0.7\), and probability of failure \(q = 0.3\). We need to find the probability of having at most three failures, which means calculating the probability for 0, 1, 2, and 3 failures.

Step 2: Formulate Probability Function for Failures

The probability of observing \(k\) failures in a binomial distribution can be computed using the formula: \(P(k \text{ failures}) = \binom{n}{k} \, p^{n-k} \, q^{k}\). Here, \(\binom{n}{k}\) is the binomial coefficient.

Step 3: Calculate Probability of 0 Failures

Substitute \(k=0\) in the formula: \(P(0 \text{ failures}) = \binom{5}{0} (0.7)^5 (0.3)^0 = 1 \cdot 0.16807 \cdot 1 = 0.16807\).

Step 4: Calculate Probability of 1 Failure

Substitute \(k=1\) in the formula: \(P(1 \text{ failure}) = \binom{5}{1} (0.7)^4 (0.3)^1 = 5 \cdot 0.2401 \cdot 0.3 = 0.36015\).

Step 5: Calculate Probability of 2 Failures

Substitute \(k=2\) in the formula: \(P(2 \text{ failures}) = \binom{5}{2} (0.7)^3 (0.3)^2 = 10 \cdot 0.343 \cdot 0.09 = 0.3087\).

Step 6: Calculate Probability of 3 Failures

Substitute \(k=3\) in the formula: \(P(3 \text{ failures}) = \binom{5}{3} (0.7)^2 (0.3)^3 = 10 \cdot 0.49 \cdot 0.027 = 0.1323\).

Step 7: Add Probabilities for At Most 3 Failures

Finally, add the probabilities for 0, 1, 2, and 3 failures to find the probability of at most 3 failures: \(P(k \leq 3) = 0.16807 + 0.36015 + 0.3087 + 0.1323 = 0.96922\).

Key Concepts

Understanding Probability Theory

Probability theory is the mathematical study of randomness and uncertainty. It allows us to quantify the likelihood of events occurring. This is crucial in predicting outcomes in a range of disciplines like finance, science, and, as in our exercise, binomial experiments. In simpler terms, probability helps us calculate how likely something is to happen.
For example, the probability of getting heads when flipping a coin is 0.5 because there are two possible outcomes, heads and tails, and both are equally likely.

• Probability values range from 0 to 1.
• A probability of 0 means the event cannot happen.
• A probability of 1 means the event is certain to happen.

In the binomial distribution exercise, probability theory helps us calculate the likelihood of experiencing a certain number of failures in five trials.

The Binomial Experiment

A binomial experiment is a specific type of probability experiment with distinct characteristics. First, it consists of a fixed number of trials or repeatable processes. Each trial can result in only one of two possible outcomes: success or failure.

• It has a consistent probability of success across each trial.
• All trials are independent, meaning the outcome of one trial doesn't affect another.

In our exercise, the trials (labelled as a binomial experiment) have five repetitions with a success probability of 0.7 and failure probability of 0.3. The aim is to find the probability of encountering at most three failures. The binomial nature of this task simplifies calculating the probabilities of interest.

Probability of Success in a Trial

The probability of success, noted as \( p \), is a key part of the binomial distribution. It signifies the chance of a successful outcome in a single trial of the experiment. In our scenario, the probability of success (i.e., not encountering a failure) is set at \( p = 0.7 \).
Knowing this allows us to define the probability of failure, \( q \), as \( 1 - p \). Therefore, if success has a probability of 0.7, failure is thus \( q = 0.3 \).
Understanding \( p \) plays a pivotal role in estimating the probability of different outcomes across multiple trials. Specifically, it's used in calculating the likelihood of a certain number of successes (or failures) using the binomial probability formula. It's important to always remember that higher \( p \) makes it more likely for outcomes with more successes to occur.

The Role of the Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{k} \), is a fundamental part of calculating binomial probabilities. It essentially answers "how many ways can we choose \( k \) successes out of \( n \) trials?" Often referred to as "combinations," it decides the number of different ways a specific number of successes can happen.
Mathematically, it is expressed as: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \( n! \) (n factorial) means multiplying all whole numbers from 1 to \( n \).
For our exercise, we used the binomial coefficient within the probability function to determine the probability of having 0, 1, 2, or 3 failures in five trials. Together with \( p \) and \( q \), it provides the full picture for the likelihood of various outcomes in the binomial experiment.