Class CBSE Class 12 Mathematics Probability Q #1508
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.
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Detailed Solution

Step 1: Define Events

Let $E_1$ be the event of choosing Bag I and $E_2$ be the event of choosing Bag II. Let $R$ be the event of drawing a red ball.

Step 2: Calculate Probabilities of Selecting Bags

A die is thrown. Bag I is chosen if the number is less than 3 (i.e., 1 or 2). Thus, $P(E_1) = \frac{2}{6} = \frac{1}{3}$. Bag II is chosen otherwise (i.e., 3, 4, 5, or 6). Thus, $P(E_2) = \frac{4}{6} = \frac{2}{3}$.

Step 3: Calculate Conditional Probabilities

Bag I has 3 red and 4 white balls (Total 7). So, $P(R|E_1) = \frac{3}{7}$. Bag II has 8 red and 6 white balls (Total 14). So, $P(R|E_2) = \frac{8}{14} = \frac{4}{7}$.

Step 4: Apply Law of Total Probability

The total probability of drawing a red ball is given by: $$P(R) = P(E_1) \cdot P(R|E_1) + P(E_2) \cdot P(R|E_2)$$ $$P(R) = \left(\frac{1}{3} \cdot \frac{3}{7}\right) + \left(\frac{2}{3} \cdot \frac{4}{7}\right)$$ $$P(R) = \frac{3}{21} + \frac{8}{21} = \frac{11}{21}$$

Final Answer: 11/21

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must identify the appropriate probability theorem (Law of Total Probability) and apply it to a multi-stage experiment.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step algorithmic approach to calculate conditional probabilities and aggregate them using the total probability formula.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This problem tests the student's ability to model real-world scenarios using probability distributions and conditional logic, which is a core requirement of the Probability chapter.

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