Q1470: 10 identical blocks are marked with '0' on two of them, '1' ...

Class CBSE Class 12 Mathematics Probability Q #1470

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2 Marks 2025 AISSCE(Board Exam) VSA

10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.

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AI Tutor Explanation

Detailed Solution

Step 1: Define the Random Variable

Let $X$ be the random variable representing the number written on a randomly selected block. The possible values of $X$ are 0, 1, 2, and 3.

Step 2: Calculate Probabilities

We need to find the probability of each value of $X$. There are a total of 10 blocks.

Step 3: Probability of X=0

There are 2 blocks marked with '0'. So, the probability of selecting a block with '0' is: $$P(X=0) = \frac{2}{10} = \frac{1}{5}$$

Step 4: Probability of X=1

There are 3 blocks marked with '1'. So, the probability of selecting a block with '1' is: $$P(X=1) = \frac{3}{10}$$

Step 5: Probability of X=2

There are 4 blocks marked with '2'. So, the probability of selecting a block with '2' is: $$P(X=2) = \frac{4}{10} = \frac{2}{5}$$

Step 6: Probability of X=3

There is 1 block marked with '3'. So, the probability of selecting a block with '3' is: $$P(X=3) = \frac{1}{10}$$

Step 7: Probability Distribution

The probability distribution of $X$ is given by: $X = 0, P(X=0) = \frac{1}{5}$ $X = 1, P(X=1) = \frac{3}{10}$ $X = 2, P(X=2) = \frac{2}{5}$ $X = 3, P(X=3) = \frac{1}{10}$

Step 8: Calculate the Mean

The mean (expected value) of $X$ is given by: $$E(X) = \sum x_i P(X=x_i)$$ $$E(X) = 0 \cdot \frac{1}{5} + 1 \cdot \frac{3}{10} + 2 \cdot \frac{2}{5} + 3 \cdot \frac{1}{10}$$ $$E(X) = 0 + \frac{3}{10} + \frac{4}{5} + \frac{3}{10}$$ $$E(X) = \frac{3}{10} + \frac{8}{10} + \frac{3}{10} = \frac{14}{10} = \frac{7}{5} = 1.4$$

Final Answer: The probability distribution is: $P(X=0) = \frac{1}{5}, P(X=1) = \frac{3}{10}, P(X=2) = \frac{2}{5}, P(X=3) = \frac{1}{10}$. The mean is 1.4.

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because the student needs to recall the formula for calculating the mean of a probability distribution and apply it to the given data.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concept of probability distribution and how to calculate the mean of a discrete random variable.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of probability distributions and their means, which is a core concept in the probability chapter.