Question

The probability that a student will pass the final examination in both English and Hindi is $0.5$ and the probability of passing neither is $0.1$. If the probability of passing the English examination is $0.75$, what is the probability of passing the Hindi examination?

Answer 1

Let $A$ and $B$ be the events of passing English and Hindi examinations respectively.
Accordingly $P(A$ and $B)=0.5,P($not A and not $B)=0.1,$ i.e., $P(A^{\prime }\cap B^{\prime })=0.1$
$P\left(A\right)=0.75$
Now ${(A\cup B)}^{\prime }=(A^{\prime }\cap B^{\prime })$ [De Morgan's law]
$\therefore P{(A\cup B)}^{\prime }=P(A^{\prime }\cap B^{\prime })=0.1$
$\Rightarrow P(A\cup B)=1-P{(A\cup B)}^{\prime }=1-0.1=0.9$
We know that $P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P(A$ and $B)$
$\therefore 0.9=0.75+P\left(B\right)-0.5$
$\Rightarrow P\left(B\right)=0.9-0.75+0.5$
$\Rightarrow P\left(B\right)=0.65$
Thus the probability of passing the Hindi examination is $0.65$