Calculating the Probability of Most Failures in a Random Sample of Students

When analyzing a sample of students, it's essential to understand the probability of various outcomes. This article focuses on the probability that a random sample of 10 students from a class will include most failures based on the given data. We will use probability concepts, specifically the binomial distribution, to find the answer.

The Context and Parameters

In a particular examination, it was observed that 60 out of 100 students passed. This gives us a pass rate of 60%. Conversely, the failure rate is 40%. We need to calculate the probability that a random sample of 10 students will include 4 or fewer failures. To achieve this, we will use the properties of the binomial distribution.

Binomial Distribution and Parameters

The scenario can be modeled using a binomial distribution since each student's success (passing or failing) is an independent event. The parameters for the binomial distribution are:

n 10 (the sample size) p 0.6 (the probability of success)

Let X be the number of passes in a random sample of 10 students. We need to find the probability that X is 4 or fewer, i.e., PX ≤ 4.

Calculating the Probability

To calculate PX ≤ 4, we need to sum the probabilities of X 0, X 1, X 2, X 3, and X 4. The formula for the binomial probability is given by:

P(X k) C(n, k) * p^k * (1-p)^(n-k)

Where C(n, k) is the number of combinations of n items taken k at a time.

Step-by-Step Calculation

P(X 0): P(X 1): P(X 2): P(X 3): P(X 4):

The probabilities can be calculated as follows:

P(X 0) C(10, 0) * 0.6^0 * 0.4^10 0.0001048576 P(X 1) C(10, 1) * 0.6^1 * 0.4^9 0.00262144 P(X 2) C(10, 2) * 0.6^2 * 0.4^8 0.014155776 P(X 3) C(10, 3) * 0.6^3 * 0.4^7 0.049129312 P(X 4) C(10, 4) * 0.6^4 * 0.4^6 0.111476736

The sum of these probabilities is:

PX ≤ 4 P(X 0) P(X 1) P(X 2) P(X 3) P(X 4)

Summing these values gives:

PX ≤ 4 0.0001048576 0.00262144 0.014155776 0.049129312 0.111476736 0.1662321126

Thus, the probability that a random sample of 10 students will include 4 or fewer failures is approximately 0.16623.

Additional Insights

It is also useful to understand the breakdown of the results from the examination. For two different ways of calculating the pass and fail percentages, we have:

1. 40% passed:
24 out of 60 students passed.
100/24 * 20 40.

2. 60% failed:
36 out of 60 students failed.
100/36 * 9 60.

These calculations demonstrate how the percentages can be derived based on the given data, providing a clear understanding of the distribution of grades.

Conclusion

In conclusion, the probability of including most failures in a random sample of 10 students, given a pass rate of 60%, is approximately 16.623%. This calculation is based on the binomial distribution and provides valuable insights into the statistical analysis of student performance in an examination.

Understanding these concepts is crucial for educators, statisticians, and anyone analyzing student performance data. By applying these principles, we can make more informed decisions and derive meaningful insights from educational data.