Case-Study 3: Read the following passage and answer the questions given below

In pre-board examination of class XII, commerce stream with Economics and Mathematics of a particular school, 50% of the students failed in Economics, 35% failed in Mathematics and 25% failed in both Economics and Mathematics. A student in selected at random from the class.

(i) Find the probability that the selected student has failed in Economics, if it is known that he has failed in Mathematics.

(ii) Find the probability that the selected student has failed in Mathematics, if it is known that he has failed in Economics.

(i) Let,

E =

 student has failed in Economics, M student has failed in Mathematics $\therefore P\left(E\right)=\frac{50}{100}=\frac{1}{2},\mathrm{P}\left(\mathrm{M}\right)=\frac{35}{100}=\frac{7}{20}$

and $\mathrm{P}(\mathrm{E}\cap \mathrm{M})=\frac{25}{100}=\frac{1}{4}$

The probability that the selected student has failed in Economics if it is known that he has failed in Mathematics. Required probability $=P\left(\frac{E}{M}\right)$

$=\frac{P(E\cap M)}{P\left(M\right)}=\frac{\frac{1}{4}}{\frac{7}{20}}=\frac{1}{4}\times \frac{20}{7}=\frac{5}{7}$

(ii) Now,

The probability that the selected student has failed in Mathematics if it is known that he has failed in Economics.

Required probability $=\mathrm{P}(\mathrm{M}/\mathrm{E})$

$=\frac{P(M\cap E)}{P\left(E\right)}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$

Therefore,

(i) The probability that the selected student has failed in Economics, if it is known that he has failed in Mathematics, it follows

$P(E\mid M)=\frac{5}{7}$

(ii) The probability that the selected student has failed in Mathematics, if it is known that he has failed in Economics, it follows

$P(M\mid E)=\frac{1}{2}$