Class CBSE Class 12 Mathematics Probability Q #912
COMPETENCY BASED
APPLY
3 Marks 2023 SA
There are two coins. One of them is a biased coin such that P (head): P (tail) is 1:3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.
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Step-by-Step Solution

Let B be the event that the biased coin is selected, and F be the event that the fair coin is selected. Let H be the event that the coin shows heads.

We are given:

  • P(H|B) = 1/4 (Probability of getting heads given the biased coin is selected)
  • P(H|F) = 1/2 (Probability of getting heads given the fair coin is selected)
  • P(B) = 1/2 (Probability of selecting the biased coin)
  • P(F) = 1/2 (Probability of selecting the fair coin)

We want to find P(B|H), the probability that the coin is biased given that it showed heads. We can use Bayes' Theorem:

P(B|H) = [P(H|B) * P(B)] / [P(H|B) * P(B) + P(H|F) * P(F)]

Substitute the given values:

P(B|H) = [(1/4) * (1/2)] / [(1/4) * (1/2) + (1/2) * (1/2)]

P(B|H) = (1/8) / (1/8 + 1/4)

P(B|H) = (1/8) / (1/8 + 2/8)

P(B|H) = (1/8) / (3/8)

P(B|H) = 1/3

Correct Answer: 1/3

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of conditional probability and Bayes' theorem to solve the problem. They must use the given information to calculate the required probability.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure (Bayes' Theorem) to calculate the conditional probability. The student needs to apply the formula correctly and perform the necessary calculations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of probability to a real-world scenario involving biased and fair coins, requiring them to use Bayes' theorem.

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