Class CBSE Class 12 Mathematics Probability Q #910
KNOWLEDGE BASED
APPLY
3 Marks 2023 SA
31. From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.
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Step-by-Step Solution

  1. Define the random variable:

    Let X be the number of defective bulbs in the sample of 2.

    X can take values 0, 1, or 2.

  2. Calculate the probabilities:

    Total number of bulbs = 30

    Number of defective bulbs = 6

    Number of non-defective bulbs = 30 - 6 = 24

    Since the bulbs are drawn with replacement, the probabilities remain constant for each draw.

    1. P(X = 0): Probability of getting 0 defective bulbs (both non-defective)

      P(X = 0) = (24/30) * (24/30) = (4/5) * (4/5) = 16/25

    2. P(X = 1): Probability of getting 1 defective bulb

      This can happen in two ways: Defective then Non-defective OR Non-defective then Defective

      P(X = 1) = (6/30) * (24/30) + (24/30) * (6/30) = 2 * (1/5) * (4/5) = 8/25

    3. P(X = 2): Probability of getting 2 defective bulbs

      P(X = 2) = (6/30) * (6/30) = (1/5) * (1/5) = 1/25

  3. Probability Distribution:

    The probability distribution of X is:

    X P(X)
    0 16/25
    1 8/25
    2 1/25

  4. Calculate the mean:

    Mean (μ) = Σ [X * P(X)]

    μ = (0 * 16/25) + (1 * 8/25) + (2 * 1/25) = 0 + 8/25 + 2/25 = 10/25 = 2/5 = 0.4

Correct Answer: 0.4

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of probability distribution and mean to solve the given problem. They must use the formulas and methods learned in class to calculate the probabilities and the mean.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to calculate the probability distribution and then use that distribution to find the mean. This involves applying formulas and following a set of steps.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of probability distributions, a core concept covered in the textbook.

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