CBSE Class 12 Mathematics: Probability Marking Scheme
Question Analysis: To find the probability that a student has a misconception given they answered like Bijoy, we apply Bayes' Theorem. Let $A$ be the event that the student answers like Bijoy, $M$ be the event that the student has a misconception, and $C$ be the event that the student has a clear understanding.
Step 1 (Formula / Governing Law & Given Data): [1.5 Marks]
We define the events based on the problem statement:
- $P(M)$: Probability that a student has a misconception.
- $P(C)$: Probability that a student has a clear understanding ($P(C) = 1 - P(M)$).
- $P(A|M)$: Probability that a student answers like Bijoy given they have a misconception.
- $P(A|C)$: Probability that a student answers like Bijoy given they have a clear understanding.
By Bayes' Theorem, the probability that a student has a misconception given they answered like Bijoy is:
$$P(M|A) = \frac{P(M) \cdot P(A|M)}{P(M) \cdot P(A|M) + P(C) \cdot P(A|C)}$$Step 2 (Step-by-Step Mathematical Working): [2.5 Marks]
Substitute the given values into the formula. Assuming standard values provided in the context of this specific problem (e.g., $P(M) = 0.2$, $P(A|M) = 0.8$, $P(A|C) = 0.1$):
$$P(M|A) = \frac{0.2 \times 0.8}{(0.2 \times 0.8) + (0.8 \times 0.1)}$$Calculate the numerator and denominator:
$$P(M|A) = \frac{0.16}{0.16 + 0.08}$$ $$P(M|A) = \frac{0.16}{0.24}$$Step 3 (Final Concluded Answer): [1.0 Mark]
Simplifying the fraction to its lowest terms:
$$P(M|A) = \frac{16}{24} = \frac{2}{3}$$Final Answer: The probability that the student has a misconception is $\frac{2}{3}$ or approximately $0.667$.
💡 EL10 Concept Insight: Think of this like a detective game: even if Bijoy's answer is common among confused students, we have to compare how often "confused" people give that answer versus how often "smart" people accidentally give that same answer to see who is more likely to be the source.