In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\cfrac{3}{4}$ be the probability that he knows the answer and $\cfrac{1}{4}$ the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\cfrac{1}{4}$ . What is the probability that the student knows the answer given that he answered it correctly?
.: Let ${E_1}$ : `the student knows the answer'
And ${E_2}$ : `the student guesses the answer'
,
$\Rightarrow$ $P\left( {{E_1}} \right) = \cfrac{3}{4}$
and $P\left( {{E_2}} \right) = \cfrac{1}{4}$
Let $A$ : 'the answer is correct',
then $P\left( {A|{E_1}} \right) = 1$ and
$P\left( {A|{E_2}} \right) = \cfrac{1}{4}$
( when the student knows the answer,
it is a sure event that the answer is correct)
Hence, the required probability is
$P\left( {{E_1}|A} \right) = \cfrac{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right)}}{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right) + P\left( {A|{E_2}} \right)P\left( {{E_2}} \right)}}$
$= \cfrac{{1 \times \cfrac{3}{4}}}{{1 \times \cfrac{3}{4} + \cfrac{1}{4} \times \cfrac{1}{4}}} = \cfrac{{\cfrac{3}{4}}}{{\cfrac{3}{4} + \cfrac{1}{{16}}}}$
$= \cfrac{{\cfrac{3}{4}}}{{\cfrac{{12 + 1}}{{16}}}} = \cfrac{3}{4} \times \cfrac{{16}}{{13}} = \cfrac{{12}}{{13}}$
NCERT & Exemplar solution for CBSE Class 12 Mathematics, Probability. Curated by Sachin Sharma. Free for all students.