Key Concepts and Formulas

• Complement Rule of Probability: If $E$ is an event, then its complement $\overline{E}$ (the event that $E$ does not occur) has a probability $P(\overline{E}) = 1 - P(E)$.
• Probability of Independent Events: If two events $A$ and $B$ are independent, the probability that both $A$ and $B$ occur is the product of their individual probabilities: $P(A \cap B) = P(A) \times P(B)$. This extends to any number of independent events.
• Probability of "At Least One" Event: For a set of events, the probability that "at least one" of them occurs is equal to $1$ minus the probability that "none" of them occur. That is, $P(\text{at least one}) = 1 - P(\text{none})$.

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Step-by-Step Solution

Step 1: Define Events and State Given Probabilities We begin by clearly defining the events associated with each student solving the problem and listing their given probabilities.

• Let $A$ be the event that student A solves the problem.
• Let $B$ be the event that student B solves the problem.
• Let $C$ be the event that student C solves the problem.

We are provided with their respective probabilities of solving the problem:

• Probability that A solves the problem, $P(A) = \frac{1}{2}$
• Probability that B solves the problem, $P(B) = \frac{1}{3}$
• Probability that C solves the problem, $P(C) = \frac{1}{4}$

Explanation: Clearly defining events and probabilities is the foundational step for any probability problem, ensuring clarity and accuracy throughout the solution.

Step 2: Calculate Probabilities of Not Solving the Problem The problem asks for the probability that the problem is solved, which means at least one student solves it. Using the complement rule, it's easier to calculate the probability that none of them solve it. For this, we first need the probabilities that each student fails to solve the problem.

• Probability that A does not solve the problem, $P(\overline{A})$: $P(\overline{A}) = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}$
• Probability that B does not solve the problem, $P(\overline{B})$: $P(\overline{B}) = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3}$
• Probability that C does not solve the problem, $P(\overline{C})$: $P(\overline{C}) = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4}$

Explanation: We calculate these probabilities because our strategy involves finding the probability that none of the students solve the problem, which directly uses these individual "failure" probabilities. This is an application of the complement rule.

Step 3: Determine the Probability that None of the Students Solve the Problem In typical competitive exam problems of this nature, the attempts of different individuals are assumed to be independent unless explicitly stated otherwise. Since the events $A, B, C$ are independent, their complements $\overline{A}, \overline{B}, \overline{C}$ are also independent. The event that "none of them solve the problem" means that A fails AND B fails AND C fails. Therefore, the probability that none of them solve the problem is the product of their individual probabilities of not solving it: $P(\text{none solve}) = P(\overline{A} \cap \overline{B} \cap \overline{C}) = P(\overline{A}) \times P(\overline{B}) \times P(\overline{C})$ Substituting the values calculated in Step 2: $P(\text{none solve}) = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}$ $P(\text{none solve}) = \frac{1 \times 2 \times 3}{2 \times 3 \times 4} = \frac{6}{24} = \frac{1}{4}$

Explanation: We multiply the probabilities because the events of each student failing are independent. If these events were dependent, a simple multiplication would not be a valid approach.

Step 4: Calculate the Probability that the Problem is Solved The event "the problem is solved" implies that at least one of the students A, B, or C solves it. This is the complement of the event that none of them solve it. Using the complement rule: $P(\text{problem is solved}) = 1 - P(\text{none solve})$ Substituting the probability calculated in Step 3: $P(\text{problem is solved}) = 1 - \frac{1}{4}$ $P(\text{problem is solved}) = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}$

Explanation: This is the final step, applying the fundamental complement rule to find the desired probability efficiently. The event "problem is solved" is the union of events A, B, and C, and its complement is the intersection of their complements.

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Common Mistakes & Tips

• Recognizing "At Least One": The phrase "at least one" in probability problems is a strong indicator that using the complement rule (calculating $1 - P(\text{none})$) will likely be the most straightforward and least error-prone method.
• Assumption of Independence: Always be mindful of the independence assumption. In problems involving different people performing tasks, independence is typically assumed unless dependency is explicitly stated or implied by the context.
• Accuracy in Complements: Double-check your calculations for the probabilities of the complement events ($P(\overline{A})$, $P(\overline{B})$, $P(\overline{C})$). A simple arithmetic error here will lead to an incorrect final answer.

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Summary

This problem effectively demonstrates the utility of the complement rule in probability, especially when dealing with "at least one" scenarios involving independent events. By first calculating the probability that none of the students solve the problem (which involves multiplying their individual failure probabilities due to independence), we can then subtract this from 1 to find the probability that at least one student solves it. This method is generally more efficient and less complex than using the inclusion-exclusion principle for three events. The final result indicates a high chance of the problem being solved.

The final answer is $\boxed{\frac{3}{4}}$ which corresponds to option (A).