A student answers a multiple choice question with 5 alternatives,… - Vidyadip

MCQ

A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is $p, 0< p< 1$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly,

A
$\frac{3 p}{4 p+3}$
B
$\frac{5 p}{3 p+2}$
✓
$\frac{5 p}{4 p+1}$
D
$\frac{4 p}{3 p+1}$

✓

Answer

Correct option: C.

$\frac{5 p}{4 p+1}$

(C)
$K = He$ knows the answers, $NK = He$ randomly ticks the answers, $C = He$ is correct
$P \left(\frac{ K }{ C }\right)=\frac{ P ( K ) \cdot P \left(\frac{ C }{ K }\right)}{ P ( K ) \cdot P \left(\frac{ C }{ K }\right)+ P ( NK ) \cdot P \left(\frac{ C }{ NK }\right)}$
$=\frac{p \times 1}{p \times 1+(1-p) \times \frac{1}{5}}=\frac{5 p}{4 p+1}$

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