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. probability of independent events cbse 12th maths old board exam question paper 2025 2026 independent events success failure type
Given:
P(Success) = 90% = 0.9 = 9/10
P(Failure) = 1 − 0.9 = 0.1 = 1/10
let S debite Success,
F denote Failure
Probability that exactly one surgery is successful
possibilities SFF, FSF, FFS
P(SFF) = 0.9 × 0.1 × 0.1 = 0.009
P(FSF) = 0.1 × 0.9 × 0.1 = 0.009
P(FFS) = 0.1 × 0.1 × 0.9 = 0.009
P(exactly one auccess) = 0.009 + 0.009 + 0.009 = 0.027 = 27/1000
Probability that at most two surgeries are successful
At most two successes means 0 or 1 or 2 successes
so
Probability that at most two surgeries are successful
= 1 − P(all three successful)
= 1 - P[ SSS]
= 1 - [0.9 × 0.9 × 0.9]
= 1 - 0.729
= 0.271
= 271/1000
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