CBSE Class 12 Mathematics Probability Q #1802
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that: (i) exactly one surgery is successful, (ii) at most two surgeries are successful.
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Detailed Solution

Step 1: Identify the Distribution

This is a Binomial Distribution problem where $n = 3$ (number of surgeries), $p = 0.9$ (probability of success), and $q = 1 - p = 0.1$ (probability of failure). The probability of $x$ successes is given by $P(X=x) = \binom{n}{x} p^x q^{n-x}$.

Step 2: Calculate exactly one success

We need to find $P(X=1)$. $$P(X=1) = \binom{3}{1} (0.9)^1 (0.1)^2$$ $$P(X=1) = 3 \times 0.9 \times 0.01 = 0.027$$

Step 3: Calculate at most two successes

We need to find $P(X \le 2)$, which is $1 - P(X=3)$. $$P(X=3) = \binom{3}{3} (0.9)^3 (0.1)^0 = 1 \times 0.729 \times 1 = 0.729$$ $$P(X \le 2) = 1 - 0.729 = 0.271$$

Final Answer: (i) 0.027, (ii) 0.271

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must identify the appropriate probability distribution model (Binomial) and apply the formula to a real-world scenario.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the execution of a specific mathematical algorithm (Binomial probability formula) to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question aligns with the NEP 2020 focus on applying theoretical probability concepts to practical, data-driven scenarios.

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