Class CBSE Class 12 Mathematics Probability Q #1805
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.
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Detailed Solution

Step 1: Define the Sample Space

Since two cards are drawn with replacement from 6 cards (numbered 1 to 6), the total number of outcomes is $6 \times 6 = 36$.

Step 2: Analyze Event A

Event A is the sum of numbers being 10. The possible pairs are (4, 6), (5, 5), and (6, 4). Thus, $P(A) = \frac{3}{36} = \frac{1}{12}$.

Step 3: Analyze Event B

Event B is getting a number other than 4 on the first card. The first card can be 1, 2, 3, 5, or 6 (5 options). The second card can be any of the 6 numbers. Thus, $n(B) = 5 \times 6 = 30$. $P(B) = \frac{30}{36} = \frac{5}{6}$.

Step 4: Calculate P(A and B)

Event (A and B) means the sum is 10 AND the first card is not 4. From the pairs in A {(4, 6), (5, 5), (6, 4)}, only (5, 5) and (6, 4) satisfy the condition of the first card not being 4. Thus, $P(A \cap B) = \frac{2}{36} = \frac{1}{18}$.

Step 5: Check for Independence

Events are independent if $P(A \cap B) = P(A) \times P(B)$. Here, $P(A) \times P(B) = \frac{1}{12} \times \frac{5}{6} = \frac{5}{72}$. Since $\frac{1}{18} \neq \frac{5}{72}$, the events are not independent.

Final Answer: P(A and B) = 1/18; Events are not independent.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate the verbal description of events into set-theoretic notation and apply the definition of independent events.
Knowledge Dimension: PROCEDURAL
Justification: The problem requires a step-by-step algorithmic approach to calculate probabilities and verify a mathematical condition.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the core understanding of Probability (Chapter 13) beyond rote memorization, requiring logical deduction of sample space subsets.

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