Understanding Binomial Distribution: A Case Study of Exam Success Rates

As a professional SEO specialist at Google, it's crucial to understand not only the intricacies of search engine optimization but also to have a broad range of knowledge in mathematics and statistics. One such interesting calculation that can be applied to real-world scenarios is understanding binomial distribution in the context of success probabilities, such as an exam. This case study will explore a situation where ten students take an exam with a success probability of 0.8, and the objective is to determine the probability of all students passing the exam.

Binomial Distribution: A Brief Overview

Binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of n independent Bernoulli trials, where each trial can result in either success or failure. Each trial is identical and has the same probability of success, denoted as p. In our case, the scenario perfectly fits a binomial distribution, with each student taking the exam being an independent trial, and the probability of success being 0.8.

Theoretical Background

tThe binomial probability mass function is given by the formula: t[ P(X k) binom{n}{k} p^k (1-p)^{n-k} ] twhere: t- ( n ) is the total number of trials (students taking the exam) t- ( k ) is the number of successes (students passing the exam) t- ( p ) is the probability of success on a single trial t- ( binom{n}{k} ) is the binomial coefficient, which can be calculated as ( frac{n!}{k!(n-k)!} )

In our specific scenario, we are interested in the probability of all ten students passing the exam, i.e., ( k 10 ) and ( n 10 ). Since the problem states that the probability of success (passing) is 0.8, we can plug these values into the formula to find the desired probability.

Calculation of the Probability

Using the binomial probability formula, the probability that all ten students will pass the exam is:

[ P(X 10) binom{10}{10} 0.8^{10} (1-0.8)^{10-10} ]

The binomial coefficient ( binom{10}{10} ) is equal to 1. The term ( (1-0.8)^{10-10} ) becomes 1 because it simplifies to ( 0.2^0 ), which is also 1. Therefore, the probability simplifies to:

[ P(X 10) 1 cdot 0.8^{10} cdot 1 0.8^{10} ]

Calculating ( 0.8^{10} ) gives us:

[ 0.8^{10} 0.1073741824 ]

This value represents the probability that all ten students will pass the exam.

Applications of Binomial Distribution

Binomial distribution is widely used in various fields, including:

tQuality control in manufacturing to determine the probability of a certain number of defective products. tMedical research to predict the likelihood of success in a treatment for a specific number of patients. tFinance to calculate the probability of a given portfolio of financial instruments performing well.

By understanding the principles of binomial distribution and applying them to real-world scenarios, we can make more accurate predictions and informed decisions.

Conclusion

In summary, the probability of all ten students passing the exam, given that the probability of success is 0.8, is 0.1073741824. This example demonstrates the practical application of binomial distribution and the importance of statistical understanding in various fields.

Further Reading

To deepen your understanding of binomial distribution and its applications, you may wish to explore the following resources:

tWikipedia: Binomial Distribution tMathIsFun: Binomial Distribution tKhan Academy: Binomial Distributions