Question

The probability of solving a question by the three students A, B, C are respectively \( \frac{3}{10}, \frac{1}{5} \) and \( \frac{1}{10} \). Find the probability of solving the question.

Hint:

For problems asking for the probability of "at least one" event occurring, it's almost always simpler to calculate the probability of the complementary event ("none" of the events occurring) and subtract the result from 1.

Answer 1

Step 1: Understanding the Concept:
The phrase "solving the question" implies that at least one of the three students solves it. It is easier to calculate the probability of the complementary event, which is that none of the students solve the question, and then subtract this from 1.
Step 2: Key Formula or Approach:
Let A, B, and C be the events that students A, B, and C solve the question, respectively. We are given:
\( P(A) = \frac{3}{10} \), \( P(B) = \frac{1}{5} \), \( P(C) = \frac{1}{10} \).
The probability that the question is solved is \( P(A \cup B \cup C) \).
Using the complement rule:
\[ P(\text{question is solved}) = 1 - P(\text{question is not solved by anyone}) \] \[ P(A \cup B \cup C) = 1 - P(A' \cap B' \cap C') \] Assuming the events are independent, \( P(A' \cap B' \cap C') = P(A') \cdot P(B') \cdot P(C') \).
Step 3: Detailed Explanation or Calculation:
First, calculate the probabilities that each student fails to solve the question.
Probability that A fails: \( P(A') = 1 - P(A) = 1 - \frac{3}{10} = \frac{7}{10} \).
Probability that B fails: \( P(B') = 1 - P(B) = 1 - \frac{1}{5} = \frac{4}{5} \).
Probability that C fails: \( P(C') = 1 - P(C) = 1 - \frac{1}{10} = \frac{9}{10} \).
Next, calculate the probability that all three of them fail. Since their attempts are independent events, we multiply their individual probabilities of failure.
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') \] \[ P(A' \cap B' \cap C') = \frac{7}{10} \times \frac{4}{5} \times \frac{9}{10} = \frac{252}{500} \] Simplify the fraction:
\[ \frac{252}{500} = \frac{126}{250} = \frac{63}{125} \] Finally, the probability that the question is solved (at least one person solves it) is 1 minus the probability that no one solves it.
\[ P(\text{question is solved}) = 1 - \frac{63}{125} = \frac{125 - 63}{125} = \frac{62}{125} \] Step 4: Final Answer:
The probability of solving the question is \( \frac{62}{125} \).