Class CBSE Class 12 Mathematics Probability Q #1741
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red?
(A) $\frac{27}{125}$
(B) $\frac{8}{125}$
(C) $\frac{2}{125}$
(D) $\frac{98}{125}$
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AI Tutor Explanation

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Detailed Solution

Step 1: Identify the probability of a single event

Total number of marbles = 4 (red) + 5 (blue) + 1 (green) = 10. The probability of drawing a red marble in one trial is $P(R) = \frac{4}{10} = \frac{2}{5}$.

Step 2: Identify the probability of the complement event

The probability of not drawing a red marble in one trial is $P(R') = 1 - P(R) = 1 - \frac{2}{5} = \frac{3}{5}$.

Step 3: Calculate probability for 3 independent trials

Since the marble is replaced, the trials are independent. The probability of not drawing a red marble in 3 trials is: $$P(\text{No Red}) = \left(\frac{3}{5}\right)^3 = \frac{27}{125}$$

Step 4: Calculate the probability of at least one red marble

Using the complement rule, the probability of at least one red marble is: $$P(\text{At least one Red}) = 1 - P(\text{No Red}) = 1 - \frac{27}{125} = \frac{125 - 27}{125} = \frac{98}{125}$$

Final Answer: 98/125

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate a real-world scenario into a binomial probability model and apply the complement rule.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a specific sequence of steps: calculating individual probability, determining the complement, and applying the power rule for independent events.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the understanding of independent events and the complement rule in Probability (Chapter 13).
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