Probability
Bayes' theorem — conditional probability
MJAT_TS7_P1
Grade 12
Question:
A student answers all true-false questions. He knows some answers and guesses the rest. $P(\text{correct}|\text{guessed})=\frac{1}{2}$. Given the student's answer is correct, the probability it was guessed is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question, given that his answer is correct, is:
A) $\dfrac{1}{12}$
B) $\dfrac{1}{7}$
C) $\dfrac{7}{5}$
D) $\dfrac{12}{5}$
Step-by-Step Solution
Key Concept: Let $P(G|C)=1/6$ (given). $P(K|C)=1-P(G|C)=5/6$. But the question asks for a different conditional — the probability that the answer is correct given it was randomly chosen... From Bayes: $P(K|C)=P(C|K)P(K)/P(C)$.
Answer: **C**.
Correct Answer: C