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.
Prev Next

AI Tutor Explanation

Powered by Gemini

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

AI generated content. Review strictly for academic accuracy.

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.

More from this Chapter

SA
The probability of simultaneous occurrence of at least one of the two events $X$ and$Y$ is $a$. If the probability that exactly one of the events $X, Y$ occurs is $b$, prove that $P(X') + P(Y') = 2 – 2a + b$.
MCQ_SINGLE
If \(P(A)=\frac{1}{7}\), \(P(B)=\frac{5}{7}\) and \(P(A\cap B)=\frac{4}{7},\) then \(P(\overline{A}|B)\) is:
SA
A die with number 1 to 6 is biased such that probability of $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.
SA
A pair of dice is thrown simultaneously. If $X$ denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of $X$.
VSA
A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?
View All Questions