Introduction

In this article, we delve into the probability of solving a problem based on the contributions of two individuals, A and B. We provide a detailed analysis using the principles of probability, specifically focusing on the union of two independent events. This method is crucial in understanding how the likelihood of one event affecting another can be calculated, ensuring clarity and precision in problem-solving.

Understanding the Problem

The scenario involves two individuals, A and B, who can solve a particular problem. The probability of A solving the problem is given as 1/4, while the probability of B solving it is 1/3. Our objective is to calculate the probability that at least one of them will solve the problem.

Step-by-Step Analysis

We can use the formula for the probability of the union of two independent events to find the solution. The formula states that:

[P(A cup B) P(A) P(B) - P(A cap B)]

Here, [P(A cup B)] is the probability that at least one of [A] or [B] solves the problem, [P(A)] is the probability that [A] solves the problem, and [P(B)] is the probability that [B] solves the problem. The term [P(A cap B)] represents the probability that both [A] and [B] solve the problem.

Calculating the Intersection

Since [A] and [B] are independent, the probability that both solve the problem is given by:

[P(A cap B) P(A) cdot P(B) frac{1}{4} cdot frac{1}{3} frac{1}{12}]

Applying the Union Formula

Now, substituting the values into the union formula, we get:

[P(A cup B) frac{1}{4} frac{1}{3} - frac{1}{12}]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 4, 3, and 12 is 12. Therefore, we convert each fraction to have the denominator 12:

[frac{1}{4} frac{3}{12}] [frac{1}{3} frac{4}{12}] [frac{1}{12} frac{1}{12}]

Substituting these values into the equation:

[P(A cup B) frac{3}{12} frac{4}{12} - frac{1}{12} frac{6}{12} frac{1}{2}]

Hence, the probability that at least one of A or B will solve the problem is:

[ boxed{frac{1}{2}} ]

Alternative Method

We can also approach this problem by calculating the probability that both fail and then subtracting from one. The probability that A fails is 3/4, and the probability that B fails is 1/4. Therefore, the probability that both fail is:

[ frac{3}{4} times frac{1}{4} frac{3}{16} ]

The probability that at least one of them succeeds is:

[ 1 - frac{3}{16} frac{13}{16} ]

Breaking Down the Problem

Let's analyze the cases where only one individual or both solve the problem:

Probability that A solves and B does not:

[ frac{1}{4} times left(1 - frac{3}{4}right) frac{1}{4} times frac{1}{4} frac{1}{16} ]

Probability that B solves and A does not:

[ frac{3}{4} times left(1 - frac{1}{4}right) frac{3}{4} times frac{3}{4} frac{9}{16} ]

Probability that both solve the problem:

[ frac{1}{4} times frac{3}{4} frac{3}{16} ]

Total probability of solving the problem:

[ frac{1}{16} frac{9}{16} frac{3}{16} frac{13}{16} ]

Thus, the overall probability is verified to be [ frac{13}{16} ], confirming our previous calculations.

Conclusion

This detailed analysis demonstrates the application of probability principles to solve real-world problems. Understanding the union of independent events and calculating the probability of complementary events are key concepts that can be applied in various scenarios. By breaking down the problem into manageable parts, we can ensure a comprehensive and accurate solution.