Question

In a class, $30\%$ 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 has failed in mathematics is
A. $\dfrac{1}{{10}}$
B. $\dfrac{2}{5}$
C. $\dfrac{9}{{20}}$
D. $\dfrac{1}{3}$

Answer 1

First of all, we will find the probability of students failed in Physics, Mathematics and both subjects i.e. $P\left( P \right),P\left( M \right),P\left( {P \cap M} \right)$. For example, we have been given $30\%$ students fail in Physics, so we have $P\left( P \right) = \dfrac{{30}}{{100}}$. Then we will use conditional probability to find the probability of a student failing in English if he fails in Hindi which can be done by using the formula $P\left( {P/M} \right) = \dfrac{{P\left( {P \cap M} \right)}}{{P\left( M \right)}}$.

Complete step by step answer:
Now, in question we are given that $30\%$ students fail in Physics; $25\%$ students fail in Mathematics and $10\%$ students fail in Physics and Mathematics both.
So, from the given data the probability of student failed in Physics can be given as,
$\Rightarrow P\left( P \right) = \dfrac{{30}}{{100}}$
The probability of a student failed in Mathematics can be given as,
$\Rightarrow P\left( M \right) = \dfrac{{25}}{{100}}$
Now, it is said that $10\%$ students fail in Physics and Mathematics both, which, means we have to take the intersection of the probability of Physics and Mathematics which can be given mathematically as,
$\Rightarrow P\left( {P \cap M} \right) = \dfrac{{10}}{{100}}$
Now, in the question, we are given that a student is chosen randomly and the probability of the student failing in Physics if she fails in Mathematics. So here we have to use the conditional probability formula which can be given as,
$P\left( {A/B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}$
Now replace A with P and B with M we will get,
$\Rightarrow P\left( {P/M} \right) = \dfrac{{P\left( {P \cap M} \right)}}{{P\left( M \right)}}$
Now, substituting the values, we will get,
$\Rightarrow P\left( {P/M} \right) = \dfrac{{\dfrac{{10}}{{100}}}}{{\dfrac{{25}}{{100}}}}$
Cancel out the common factors,
$\therefore P\left( {P/M} \right) = \dfrac{2}{5}$
Thus, the probability is $\dfrac{2}{5}$.

Hence, option (B) is the correct answer.

Note: There are chances of students making mistakes in writing the formula for conditional probability i.e. instead of writing $P\left( {P/M} \right) = \dfrac{{P\left( {P \cap M} \right)}}{{P\left( M \right)}}$, in denominator student take the probability of Physics in place of Mathematics and answer gets wrong i.e. $P\left( {P/M} \right) = \dfrac{{P\left( {P \cap M} \right)}}{{P\left( P \right)}}$. Also, in numerator writing $\left( {P \cap M} \right)$ , or $\left( {M \cap P} \right)$ will not affect the answer but changing the denominator will affect the answer. So, don’t make this mistake.