Class CBSE Class 12 Mathematics Probability Q #1348
KNOWLEDGE BASED
REMEMBER
3 Marks 2024 AISSCE(Board Exam) SA
The random variable X has the following probability distribution where a and b are some constants: $P(X)$ for X=1 is 0.2, X=2 is a, X=3 is a, X=4 is 0.2, X=5 is b. If the mean $E(X)=3$, then find values of a and b and hence determine $P(X\ge3)$
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Detailed Solution

Step 1: Use the property that the sum of probabilities is 1.

Since the sum of probabilities for all possible values of a random variable must equal 1, we have: $0.2 + a + a + 0.2 + b = 1$ $2a + b = 1 - 0.4$ $2a + b = 0.6$ ...(1)

Step 2: Use the given mean to form another equation.

The mean (expected value) $E(X)$ is given by: $E(X) = \sum x \cdot P(x)$ $E(X) = 1(0.2) + 2(a) + 3(a) + 4(0.2) + 5(b) = 3$ $0.2 + 2a + 3a + 0.8 + 5b = 3$ $5a + 5b = 3 - 1$ $5a + 5b = 2$ $a + b = 0.4$ ...(2)

Step 3: Solve the system of equations.

We have two equations: $2a + b = 0.6$ ...(1) $a + b = 0.4$ ...(2) Subtract equation (2) from equation (1): $(2a + b) - (a + b) = 0.6 - 0.4$ $a = 0.2$ Substitute $a = 0.2$ into equation (2): $0.2 + b = 0.4$ $b = 0.4 - 0.2$ $b = 0.2$

Step 4: Determine the values of a and b.

So, $a = 0.2$ and $b = 0.2$.

Step 5: Calculate $P(X \ge 3)$.

We need to find $P(X \ge 3)$, which is $P(X=3) + P(X=4) + P(X=5)$. $P(X \ge 3) = a + 0.2 + b = 0.2 + 0.2 + 0.2 = 0.6$

Final Answer: a = 0.2, b = 0.2, P(X>=3) = 0.6

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the definitions of probability distribution, expected value, and applying basic algebraic manipulation to solve for unknowns.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concepts of probability distribution, expected value, and how they relate to each other. It requires applying these concepts to solve for unknown variables.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the student's understanding and application of concepts related to probability distributions and expected values, which are core topics in the syllabus.

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