Introduction

The problem presented involves a scenario where three students independently work on a problem to solve it. The probability that the first student solves it is 0.95, the second student has a 0.85 probability of solving it, and the third student has a 0.80 probability. We are interested in finding the probability that either the first or the second student solves the problem.

The Probability of At Least One Student Solving the Problem

To solve this problem, we can use the principle of complementary probability. This involves calculating the probability that the event of interest occurs, which in this case is at least one of the first two students solving the problem, by considering the complementary event—i.e., the probability that neither of the first two students solves it.

Let's denote the probabilities as follows:

PA 0.95, the probability that the first student solves the problem. PB 0.85, the probability that the second student solves the problem.

We want to find the probability that at least one of the first two students solves the problem. This can be calculated using the union formula for probabilities:

PA ∪ B PA PB - PA ∩ B

Here, PA ∩ B represents the probability that both the first and the second students solve the problem. Since the students work independently, the probability that both solve the problem is simply:

PA ∩ B PA × PB 0.95 × 0.85

Calculating PA ∩ B:

PA ∩ B 0.95 × 0.85 0.8075

Now, substituting the values into the union formula:

PA ∪ B 0.95 0.85 - 0.8075 1.8 - 0.8075 0.9925

Therefore, the probability that either the first student or the second student solves the problem is:

PA ∪ B 0.9925

Assumptions and Interpretations

It's important to recognize that this answer is slightly over 95%, which indicates that it is highly likely that at least one of the first two students will solve the problem.

The assumption that the events of each student solving the problem are independent requires careful consideration. For example, if Student 2 knew that Student 1 had a significant probability of solving the problem, it would not be reasonable to assume that Student 2's probability of solving the problem is 85% independently of whether Student 1 managed to solve it. This assumption would only hold if the problem was completely unrelated and Student 2 had no knowledge or indication of Student 1's success.

To verify that the probability of at least one of the first two students solving the problem is indeed 0.9925, we can also calculate it as follows:

1 - (1 - PA) × (1 - PB) 1 - (1 - 0.95) × (1 - 0.85) 1 - 0.05 × 0.15 1 - 0.0075 0.9925

This approach assumes that whether Student 3 solves the problem is irrelevant, focusing solely on the probability that at least one of the first two students solves it.

Conclusion

Thus, the probability that either the first student or the second student solves the problem is approximately 99.25%, assuming their efforts are completely independent.

Keywords: probability, independent events, complementary probability