Class CBSE Class 12 Mathematics Probability Q #908
COMPETENCY BASED
APPLY
5 Marks 2023 LA
(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/3. What is the probability that the student knows the answer, given that he answered it correctly? OR (b) A box contains 10 tickets, 2 of which carry a prize of ₹8 each, 5 of which carry a prize of ₹4 each, and remaining 3 carry a prize of ₹2 each. If one ticket is drawn at random, find the mean value of the prize.
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Step-by-Step Solution

(a) Let A be the event that the student knows the answer, and C be the event that the student answers correctly. We are given:

P(A) = 3/5 (probability that the student knows the answer)

P(A') = 2/5 (probability that the student guesses)

P(C|A) = 1 (probability of answering correctly given the student knows the answer)

P(C|A') = 1/3 (probability of answering correctly given the student guesses)

We want to find P(A|C), the probability that the student knows the answer given that they answered correctly. Using Bayes' Theorem:

P(A|C) = [P(C|A) * P(A)] / [P(C|A) * P(A) + P(C|A') * P(A')]

P(A|C) = [1 * (3/5)] / [1 * (3/5) + (1/3) * (2/5)]

P(A|C) = (3/5) / (3/5 + 2/15)

P(A|C) = (3/5) / (9/15 + 2/15)

P(A|C) = (3/5) / (11/15)

P(A|C) = (3/5) * (15/11)

P(A|C) = 9/11

(b) Let X be the random variable representing the value of the prize. The possible values of X are ₹8, ₹4, and ₹2. The probabilities of drawing each prize are:

P(X = 8) = 2/10 = 1/5

P(X = 4) = 5/10 = 1/2

P(X = 2) = 3/10

The mean value of the prize, E(X), is calculated as:

E(X) = Σ [x * P(x)]

E(X) = (8 * 1/5) + (4 * 1/2) + (2 * 3/10)

E(X) = 8/5 + 4/2 + 6/10

E(X) = 1.6 + 2 + 0.6

E(X) = 4.2

Correct Answer: (a) 9/11 OR (b) ₹4.2

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply Bayes' Theorem or probability concepts to solve the problem, rather than just recalling a formula.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of probability concepts like conditional probability and Bayes' theorem, rather than just recalling facts or procedures.
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, aligning with competency-based assessment principles.

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