Introduction

The process of solving a problem accurately by individuals often varies among individuals. Given the probabilities of solving a problem correctly for three students, let's dive into the calculations to find the probability that none of the students solved the problem correctly. This involves understanding the concept of independent events and how to calculate the probability of multiple events occurring.

Understanding the Probabilities

We are given the probabilities of each student solving the problem correctly:

Student A: (P_A frac{1}{5}) Student B: (P_B frac{2}{3}) Student C: (P_C frac{2}{5})

To find the probability that none of the students solved the problem correctly, we first need to determine the probability that each student did not solve the problem. This is calculated by subtracting the probability of solving from 1 for each student.

Calculating the Probability of Not Solving

The probabilities that each student did not solve the problem are:

For Student A: (P_{text{not } A} 1 - P_A 1 - frac{1}{5} frac{4}{5}) For Student B: (P_{text{not } B} 1 - P_B 1 - frac{2}{3} frac{1}{3}) For Student C: (P_{text{not } C} 1 - P_C 1 - frac{2}{5} frac{3}{5})

Calculating the Combined Probability

Since the events are independent, we multiply the probabilities of each student not solving the problem. The combined probability that none of the students solved the problem is:

(P_{text{none solved}} P_{text{not } A} times P_{text{not } B} times P_{text{not } C})

Substituting the values:

(P_{text{none solved}} left(frac{4}{5}right) times left(frac{1}{3}right) times left(frac{3}{5}right))

Calculating the product:

(P_{text{none solved}} frac{4 times 1 times 3}{5 times 3 times 5} frac{12}{75})

Simplifying the fraction:

(frac{12}{75} frac{4}{25})

Therefore, the probability that none of the students solved the problem correctly is (frac{4}{25}).

Conclusion and Further Exploration

This example demonstrates how to calculate the probability of multiple independent events not happening. It is essential to clearly define the problem and consider the probabilities of each event before performing the calculations. Understanding these concepts can help in solving more complex probability problems.