Solving the Probability Problem with Independent Events

This problem asks for the probability that a statistics problem is solved, given the independent probabilities of three students A, B, and C solving it individually.

We are given the probabilities that each student solves the problem:

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

Since the students solve the problem independently, their actions do not influence each other.

Calculating Probability of Failure

To find the probability that the problem is solved, it's easier to calculate the probability that the problem is *not* solved by any of the students and subtract this from 1. The event that the problem is not solved means that student A fails AND student B fails AND student C fails.

Let \(P(A')\) be the probability that student A does not solve the problem. Using the complement rule, \(P(A') = 1 - P(A)\).

• \(P(A') = 1 - \frac{1}{2} = \frac{1}{2}\)
• \(P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3}\)
• \(P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4}\)

Probability that None Solve the Problem

Since the students work independently, the probability that none of them solve the problem is the product of their individual probabilities of failure:

\(P(\text{None solve}) = P(A' \text{ and } B' \text{ and } C') = P(A') \times P(B') \times P(C')\)

Substituting the values we calculated:

\(P(\text{None solve}) = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}\)

Multiplying the fractions:

\(P(\text{None solve}) = \frac{1 \times 2 \times 3}{2 \times 3 \times 4} = \frac{6}{24}\)

Simplifying the fraction:

\(P(\text{None solve}) = \frac{1}{4}\)

Probability of the Problem Being Solved

The event that the problem is solved is the complement of the event that none of the students solve it. Therefore, the probability that the problem is solved is:

\(P(\text{Problem is solved}) = 1 - P(\text{None solve})\)

Substituting the probability that none solve it:

\(P(\text{Problem is solved}) = 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}\)

Thus, the probability that the problem will be solved by at least one of the students is \(\frac{3}{4}\).

Revision Table: Key Probability Concepts

Additional Information on Probability Calculations

Understanding independence is crucial in probability. When events are independent, we can simply multiply their probabilities to find the likelihood of all of them occurring. This simplifies calculations significantly compared to dependent events where conditional probabilities must be considered.

The complement rule is another powerful tool. When finding the probability of "at least one" of several events happening, calculating the probability of the complementary event ("none" of the events happening) and subtracting it from 1 is often the most straightforward method, especially when dealing with independent events.

In this problem, "the problem is solved" means that either A solves it, or B solves it, or C solves it, or any combination of them solve it. Calculating the probability of each of these individual scenarios and summing them up (being careful about overlaps for combinations) would be much more complicated. The complement approach avoids this complexity entirely.