Calculating Independent Problem-Solving Probabilities: A Comprehensive Guide

In the context of problem-solving, it is important to understand the probabilities involved when multiple individuals attempt to find a solution independently. This article outlines how to calculate the probability that one or more individuals will solve a problem, but another will not. Let's delve into the scenario involving Xavier, Yvonne, and Zelda, each with their own success probabilities.

Introduction to Independent Probabilities

When three people, Xavier, Yvonne, and Zelda, attempt to solve a problem independently, their individual success probabilities are given as:

Xavier: 1/4 Yvonne: 1/2 Zelda: 5/8

The important concept here is that the events are independent, meaning the outcome of one individual solving the problem does not affect the others.

Probability Calculations

To find the probability that Xavier and Yvonne will solve the problem, but Zelda will not, we need to use the formula for independent events. The probability of an event not occurring is calculated as 1 minus the probability of it occurring.

Step-by-Step Solution

Calculate the probability that Zelda does not solve the problem:

P(Z not solving) 1 - 5/8 3/8

Multiply the probabilities of Xavier, Yvonne solving the problem, and Zelda not solving it:

P(X and Y solve, Z does not) ' 1/4 × 1/2 × 3/8 3/64

Thus, the probability that Xavier and Yvonne will solve the problem but Zelda will not is 3/64.

Other Probability Scenarios

The probability that only Xavier solves the problem is: P(XY'Z') 1/4 × 1/2 × 3/8 3/64 The probability that only Yvonne solves the problem is: P(X'YZ') (1 - 1/4) × 1/2 × 3/8 9/64 The probability that only Zelda solves the problem is: P(X'Y'Z) (1 - 1/4) × (1 - 1/2) × 5/8 15/64

Additional Calculations

The probability that only Xavier and Zelda solve the problem is: P(XYZ') 1/4 × 1/2 × 3/8 3/64 The probability that only Yvonne and Zelda solve the problem is: P(X'YZ) (1 - 1/4) × 1/2 × 5/8 15/64 The probability that no one solves the problem is: P(X'Y'Z') (1 - 1/4) × (1 - 1/2) × (1 - 5/8) 9/64 The probability that all three of them solve the problem is: P( XYZ) 1/4 × 1/2 × 5/8 5/64

The sum of all these probabilities should total 1, which is 64/64.

Conclusion

Understanding how to calculate independent probabilities is crucial in various fields, including mathematics, statistics, and decision-making. The example of Xavier, Yvonne, and Zelda provides a clear illustration of how to calculate such probabilities, ensuring that the sum of all possible outcomes equals 1.