Class CBSE Class 12 Mathematics Probability Q #1803
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.
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Detailed Solution

Step 1: Define Events and Probabilities

Let $D$ be the event that a debate competition is held, and $Q$ be the event that a quiz competition is held. Given: $P(D) = \frac{1}{3}$ and $P(Q) = \frac{2}{3}$.

Step 2: Calculate Selection Probabilities for Team A

Team A has 4 girls and 6 boys (Total 10). The probability of selecting 1 girl and 1 boy from Team A is: $$P(E|D) = \frac{\binom{4}{1} \times \binom{6}{1}}{\binom{10}{2}} = \frac{4 \times 6}{45} = \frac{24}{45} = \frac{8}{15}$$

Step 3: Calculate Selection Probabilities for Team B

Team B has 7 girls and 3 boys (Total 10). The probability of selecting 1 girl and 1 boy from Team B is: $$P(E|Q) = \frac{\binom{7}{1} \times \binom{3}{1}}{\binom{10}{2}} = \frac{7 \times 3}{45} = \frac{21}{45} = \frac{7}{15}$$

Step 4: Apply Law of Total Probability

The total probability $P(E)$ is given by: $$P(E) = P(D) \times P(E|D) + P(Q) \times P(E|Q)$$ $$P(E) = \left(\frac{1}{3} \times \frac{8}{15}\right) + \left(\frac{2}{3} \times \frac{7}{15}\right)$$ $$P(E) = \frac{8}{45} + \frac{14}{45} = \frac{22}{45}$$

Final Answer: 22/45

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to integrate the Law of Total Probability with combinatorial selection methods.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a specific sequence of steps: calculating conditional probabilities using combinations and then aggregating them using the total probability theorem.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the student's ability to model a real-world scenario using probability distributions and selection logic, aligning with the NEP 2020 focus on conceptual application.

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