Question

Mark the correct alternative in the following question:

In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics is

$\left(\mathrm{a}\right)\frac{1}{10}\left(\mathrm{b}\right)\frac{1}{3}\left(\mathrm{c}\right)\frac{2}{5}\left(\mathrm{d}\right)\frac{9}{20}$

Answer 1

$$\begin{gathered} \mathrm{Let}A\mathrm{be}\mathrm{the}\mathrm{event}\mathrm{of}\mathrm{choosing}\mathrm{a}\mathrm{student}\mathrm{failed}\mathrm{in}\mathrm{Physics}\mathrm{and} \\ B\mathrm{be}\mathrm{the}\mathrm{event}\mathrm{of}\mathrm{choosing}\mathrm{a}\mathrm{student}\mathrm{failed}\mathrm{in}\mathrm{Mathematics}. \\ \mathrm{We}\mathrm{have}, \\ \mathrm{P}\left(A\right)=30\%=\frac{30}{100}=\frac{3}{10}, \\ \mathrm{P}\left(B\right)=25\%=\frac{25}{100}=\frac{1}{4}\mathrm{and} \\ \mathrm{P}\left(A\cap B\right)=10\%=\frac{10}{100}=\frac{1}{10} \\ \mathrm{Now}, \\ \mathrm{P}\left(A|B\right)=\frac{\mathrm{P}\left(A\cap B\right)}{\mathrm{P}\left(B\right)} \\ =\frac{\left({\displaystyle \frac{1}{10}}\right)}{\left({\displaystyle \frac{1}{4}}\right)} \\ =\frac{4}{10} \\ =\frac{2}{5} \end{gathered}$$

Hence, the correct alternative is option (c).