Class CBSE Class 12 Mathematics Probability Q #1412
KNOWLEDGE BASED
REMEMBER
3 Marks 2025 AISSCE(Board Exam) SA
A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.
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Detailed Solution

Step 1: Define the Sample Space

When a coin is tossed twice, the sample space S is given by: S = {HH, HT, TH, TT}

Step 2: Define the Random Variable X

Let X be the random variable representing the number of heads minus the number of tails.

Step 3: Determine the Values of X for each outcome

For each outcome in the sample space, we calculate the value of X: - HH: 2 heads, 0 tails. X = 2 - 0 = 2 - HT: 1 head, 1 tail. X = 1 - 1 = 0 - TH: 1 head, 1 tail. X = 1 - 1 = 0 - TT: 0 heads, 2 tails. X = 0 - 2 = -2

Step 4: Determine the Probability Distribution of X

The possible values of X are -2, 0, and 2. We calculate the probabilities for each value: - P(X = -2) = P(TT) = 1/4 - P(X = 0) = P(HT) + P(TH) = 1/4 + 1/4 = 2/4 = 1/2 - P(X = 2) = P(HH) = 1/4 The probability distribution of X is: X | -2 | 0 | 2 ------- | -------- | -------- | -------- P(X) | 1/4 | 1/2 | 1/4

Step 5: Calculate the Mean of X

The mean (or expected value) of X, denoted by E(X), is calculated as: E(X) = Σ [x * P(x)] E(X) = (-2 * 1/4) + (0 * 1/2) + (2 * 1/4) E(X) = -2/4 + 0 + 2/4 E(X) = 0

Final Answer: The probability distribution is: X: -2, 0, 2 with probabilities 1/4, 1/2, 1/4 respectively. The mean is 0.

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the definition of probability distribution and the formula for calculating the mean of a random variable.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of random variables, probability distributions, and expected value (mean). It involves applying these concepts to a specific scenario (coin toss).
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the syllabus content on probability distributions and random variables. The question tests the student's understanding of basic definitions and formulas.

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