# Which one of the following is correct?

In an examination $30 \%$ of the students failed in Mathematics, $15 \%$ of the students failed in English and $10 \%$ of the students failed in both Mathematics and English. A student is chosen at random. If he failed in English then the probability that he passed in Mathematics is

$(a)$ $\frac {1} {2}.$

$(b)$ $\frac {1} {10}.$

$(c)$ $\frac {1} {3}.$

$(d)$ $\frac {7} {10}.$

My attempt $:$

Suppose if we take $100$ students as the total number of students in the class. Then out of these $100$ the number of students qualify in Mathematics is $70$ and the number of students qualify in English is $85$. Since $10$ were failed in both the subjects. So the total number of students who qualify in both the subjects is $70+85-90=65.$ So the number of students who have qualified in Mathematics but not in English is $70-65=5.$ Now the total number of students who have not qualified in English is given as $15$ and hence the required probability is $\frac {5} {15}$ which simplifies to $\frac {1} {3}.$ So according to me $(c)$ is the correct option.

Is the above reasoning correct at all? Please verify it.

Yes, correct! No need to figure out all those who passed both subjects though. You just needed to point out that out of the 15 that failed English, 10 also failed Math, so 5 out of the 15 that failed English passed Math which is $\frac{1}{3}$