CBSE Class 12 Mathematics Probability Q #1324
COMPETENCY BASED
UNDERSTAND
3 Marks 2024 AISSCE(Board Exam) SA
A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.
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Detailed Solution

Step 1: Define Events

Let $K$ be the event that the lost card is a King. Let $E$ be the event that a card drawn from the remaining cards is a King.

Step 2: Determine Prior Probabilities

We want to find $P(K|E)$, the probability that the lost card is a King given that a card drawn is a King. $P(K) = \frac{4}{52} = \frac{1}{13}$ (Probability that the lost card is a King) $P(K') = 1 - P(K) = 1 - \frac{1}{13} = \frac{12}{13}$ (Probability that the lost card is not a King)

Step 3: Determine Conditional Probabilities

$P(E|K) = \frac{3}{51}$ (Probability of drawing a King given that the lost card was a King) $P(E|K') = \frac{4}{51}$ (Probability of drawing a King given that the lost card was not a King)

Step 4: Apply Bayes' Theorem

Using Bayes' Theorem, we have: $$P(K|E) = \frac{P(E|K)P(K)}{P(E|K)P(K) + P(E|K')P(K')}$$ $$P(K|E) = \frac{\frac{3}{51} \cdot \frac{1}{13}}{\frac{3}{51} \cdot \frac{1}{13} + \frac{4}{51} \cdot \frac{12}{13}}$$ $$P(K|E) = \frac{\frac{3}{51 \cdot 13}}{\frac{3}{51 \cdot 13} + \frac{48}{51 \cdot 13}}$$ $$P(K|E) = \frac{3}{3 + 48} = \frac{3}{51} = \frac{1}{17}$$

Final Answer: 1/17

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concepts of conditional probability and Bayes' theorem to apply them to the given problem.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of the concepts of conditional probability and Bayes' theorem, rather than just recalling facts or following a specific procedure.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires application of probability concepts to a real-world scenario, testing the student's problem-solving skills.

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