Question

In a college, 70% students pass in physics, 75% pass in mathematics and 10% students fail in both. One student is chosen at random. What is the probability that:
$\left( i \right)$ He passes in physics and mathematics.
$\left( {ii} \right)$ He passes in mathematics given that he passes in Physics.
$\left( {iii} \right)$ He passes in physics given that he passes in mathematics.

Answer 1

Hint: In this particular type of question use the concept that in a set the union (total elements in the two sets) of two sets is equal to the sum of individual elements of the set and the difference of the common elements in the set, i.e. $n\left( {P \cup M} \right) = n\left( P \right) + n\left( M \right) - n\left( {P \cap M} \right)$and use the concept that the probability if he passes in physics given that he passes in mathematics is $P\left( {\dfrac{P}{M}} \right) = \dfrac{{n\left( {P \cap M} \right)}}{{n\left( M \right)}}$ so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
In a college, 70% students pass in physics, 75% pass in mathematics
Therefore, n(P) = 70%, n(M) = 75%
Let the total number of students in the class is = 100%.
Now it is given that 10% students fail in both physics and mathematics so the remaining percentage of students who can pass either in physics or in chemistry or in both = 100% - 10% = 90%.
So, $n\left( {P \cup M} \right) = 90{\text{% }}$
So as we know in a set,
$n\left( {P \cup M} \right) = n\left( P \right) + n\left( M \right) - n\left( {P \cap M} \right)$
Where, $n\left( {P \cap M} \right)$ are the number of students passed in both the subjects.
Now substitute the values we have,
\[ \Rightarrow 90{\text{% }} = 70{\text{% }} + 75{\text{% }} - n\left( {P \cap M} \right)\]
\[ \Rightarrow n\left( {P \cap M} \right) = 70{\text{% }} + 75{\text{% }} - 90{\text{% }} = 55{\text{% }}\]
Now the percentage of students who pass only in physics = 70% - 55% = 15%.
And the number of students who pass only in mathematics = 75% - 55% = 20%.
So 55% percent of students pass in both the subjects.
Now as we know that the probability is the ratio of favorable number of outcomes to the total number of outcomes.
Therefore, P = $\dfrac{{{\text{favorable number of outcomes}}}}{{{\text{total number of outcomes}}}}$
$\left( i \right)$ He passes in physics and mathematics.
So when one student chosen at random favorable number of students which are passes in both physics and chemistry = 55%
And the total number of outcomes = 100%
So the probability = $\dfrac{{55{\text{% }}}}{{100{\text{% }}}} = 0.55$
$\left( {ii} \right)$ He passes in mathematics given that he passes in Physics.
So we have to find the conditional probability i.e. P (M/P)
So, $P\left( {\dfrac{M}{P}} \right) = \dfrac{{n\left( {P \cap M} \right)}}{{n\left( P \right)}}$
Now substitute the values we have,
So, $P\left( {\dfrac{M}{P}} \right) = \dfrac{{55{\text{% }}}}{{70{\text{% }}}} = \dfrac{{11}}{{14}}$
$\left( {iii} \right)$ He passes in physics given that he passes in mathematics.
So we have to find the conditional probability i.e. P (P/M)
So, $P\left( {\dfrac{P}{M}} \right) = \dfrac{{n\left( {P \cap M} \right)}}{{n\left( M \right)}}$
Now substitute the values we have,
So, $P\left( {\dfrac{P}{M}} \right) = \dfrac{{55{\text{% }}}}{{75{\text{% }}}} = \dfrac{{11}}{{15}}$
So these are the required answers.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the definition of the probability which is stated above, so first find out the number of students who pass in both the subjects using set theory as above then find out the percentage of students who pass in individual subjects as above then apply probability formula and also the conditional probability formula we will get the required answer.