Probability of Problem Resolution Among Three Individuals

In scenarios where multiple individuals are given a problem and each has a different probability of solving it, understanding the combined probabilities is crucial for making accurate predictions. This article explores the probability that at least one of three individuals, A, B, and C, will solve a given problem, with their probabilities of solving it being 1/2, 1/3, and 1/4 respectively.

Understanding the Problem

Given the problem, let us break down the probabilities:

Probability that A can solve the problem: P(A) 1/2 Probability that A cannot solve the problem: P(A') 1/2 Probability that B can solve the problem: P(B) 1/3 Probability that B cannot solve the problem: P(B') 2/3 Probability that C can solve the problem: P(C) 1/4 Probability that C cannot solve the problem: P(C') 3/4

Probability that at Least One Solves the Problem

To determine the probability that at least one of the individuals (A, B, or C) will solve the problem, we can use the complement rule. This involves first calculating the probability that none of them solve the problem, then subtracting that from 1.

Let's start by calculating the probability that each person does not solve the problem:

P(A') 1 - P(A) 1 - 1/2 1/2 P(B') 1 - P(B) 1 - 1/3 2/3 P(C') 1 - P(C) 1 - 1/4 3/4

Now, we calculate the probability that none of them solve the problem:

P(A' ∩ B' ∩ C') P(A') × P(B') × P(C') 1/2 × 2/3 × 3/4

Let's calculate this step-by-step:

P(A' ∩ B) 1/2 × 2/3 1/3 P(A' ∩ B ∩ C') 1/3 × 3/4 1/4

Finally, the probability that at least one of them solves the problem is:

P(at least one solves) 1 - P(A' ∩ B' ∩ C') 1 - 1/4 3/4

Probability of Exactly One Individual Solving the Problem

To find the probability that exactly one of the individuals (A, B, or C) will solve the problem, we consider the following scenarios:

A solves, B and C do not solve: P(A) × P(B') × P(C') 1/2 × 2/3 × 3/4 6/24 B solves, A and C do not solve: P(A') × P(B) × P(C') 1/2 × 1/3 × 3/4 3/24 C solves, A and B do not solve: P(A') × P(B') × P(C) 1/2 × 2/3 × 1/4 2/24

The combined probability that exactly one of them will solve the problem is given by the sum of the above probabilities:

P(exactly one solves) 6/24 3/24 2/24 11/24

This detailed calculation highlights the importance of understanding and applying the rules of probability in such scenarios, providing a clear theoretical foundation for accurate predictions.