Q1458: A person is Head of two independent selection committees I a...

Class CBSE Class 12 Mathematics Probability Q #1458

COMPETENCY BASED

UNDERSTAND

3 Marks 2025 AISSCE(Board Exam) SA

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection: (i) in both committees (ii) in only one committee.

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AI Tutor Explanation

Detailed Solution

Step 1: Define probabilities

Let $P(W_1)$ be the probability of making a wrong selection in committee I, and $P(W_2)$ be the probability of making a wrong selection in committee II. We are given: $P(W_1) = 0.03$ $P(W_2) = 0.01$

Step 2: Calculate probabilities of correct selection

Let $P(C_1)$ be the probability of making a correct selection in committee I, and $P(C_2)$ be the probability of making a correct selection in committee II. Then: $P(C_1) = 1 - P(W_1) = 1 - 0.03 = 0.97$ $P(C_2) = 1 - P(W_2) = 1 - 0.01 = 0.99$

Step 3: Calculate probability of correct selection in both committees

Since the committees are independent, the probability of making the correct selection in both committees is: $P(C_1 \cap C_2) = P(C_1) \times P(C_2) = 0.97 \times 0.99 = 0.9603$

Step 4: Calculate probability of correct selection in only one committee

The probability of making the correct selection in only one committee can be calculated as the sum of the probabilities of making a correct selection in committee I and a wrong selection in committee II, and making a wrong selection in committee I and a correct selection in committee II. $P(\text{Only one correct}) = P(C_1 \cap W_2) + P(W_1 \cap C_2) = P(C_1)P(W_2) + P(W_1)P(C_2)$ $P(\text{Only one correct}) = (0.97)(0.01) + (0.03)(0.99) = 0.0097 + 0.0297 = 0.0394$

Final Answer: (i) 0.9603 (ii) 0.0394

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concepts of probability, independent events, and how to calculate probabilities of complementary events to solve the problem.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure to calculate the probabilities, involving understanding of independent events and complementary probabilities.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply probability concepts to a real-world scenario involving selection committees, requiring them to understand and apply the rules of probability for independent events.