Class CBSE Class 12 Mathematics Probability Q #692
KNOWLEDGE BASED
REMEMBER
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:
(A) \(\frac{124}{125}\)
(B) \(\frac{1}{125}\)
(C) \(\frac{61}{125}\)
(D) \(\frac{64}{125}\)
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Correct Answer: C

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Detailed Solution

Step 1: Calculate the total number of pens.

The total number of pens in the box is $4 \text{ (green)} + 8 \text{ (blue)} + 3 \text{ (red)} = 15$.

Step 2: Calculate the probability of picking a red pen.

The probability of picking a red pen in a single draw is $\frac{3}{15} = \frac{1}{5}$.

Step 3: Calculate the probability of not picking a red pen.

The probability of not picking a red pen in a single draw is $1 - \frac{1}{5} = \frac{4}{5}$.

Step 4: Calculate the probability of not picking a red pen in 3 draws.

Since the pen is replaced each time, the draws are independent. Therefore, the probability of not picking a red pen in 3 draws is $\left(\frac{4}{5}\right)^3 = \frac{64}{125}$.

Step 5: Calculate the probability of picking at least one red pen in 3 draws.

The probability of picking at least one red pen is the complement of not picking any red pens in 3 draws. Therefore, the probability is $1 - \frac{64}{125} = \frac{125 - 64}{125} = \frac{61}{125}$.

Final Answer: \(\frac{61}{125}\)

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the basic concepts of probability and applying the complement rule.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of concepts like independent events and complementary probability.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the textbook content on probability, specifically independent events and the use of complementary probability.
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