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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability2 Marks
Question
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac {1}{4}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$. What is the probability that the student knows the answer given that he answered it correctly?
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Answer

Let E1 : the student knows the answer, then, P(E1) = $ \frac{3}{4}$
E2 : the student guesses the answer, then, P(E2) = $ \frac{1}{4}$
Let A: the answer is correct.
$P(\frac {A}{{E_1}}) $ = 1, $P(\frac {A}{{E_2}}) = \frac{1}{4}$
Hence, by Baye's theorm, we have,
$P(\frac {E_1}{A}) = \frac{{P({E_1})P(A/{E_1})}}{{P({E_1})P(A/{E_1}) + P({E_2})P(A/{E_2})}}$
$ = \frac{{1 \times \frac{3}{4}}}{{1 \times \frac{3}{4} + \frac{1}{4} \times \frac{1}{4}}} = \frac{{12}}{{13}}$

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