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 $E$ be the event that student passing the English examination.
Let $H$ be the event that student passing the Hindi examination.
Let $(E\cap H)$ be the event that student passing both examination.
Let $(E\text{'}\cap H\text{'})$ be the event that student not passing both examination
$P\left(E\right)=0.75,P(E\cap H)=0.5$ and $P(E^{\prime }\cap H^{\prime })=0.1....\left(i\right)$
$P(E^{\prime }\cap H^{\prime })=P(E\cup H{)}^{\prime }$ $($By Demorgan law$)$
$P(E\cup H{)}^{\prime }+P(E\cup H)=1$
$P(E\cup H)=1-P(E\cup H{)}^{\prime }$
$P(E\cup H)=1-0.1=0.9.....\left(ii\right)$

As we know that
$P(E\cup H)=P\left(E\right)+P\left(H\right)-P(E\cap H)....\left(iii\right)$

Substituting values $\left(i\right)$ and $\left(ii\right)$ in $\left(iii\right)$
$0.9=0.75+P\left(H\right)-0.5$
$P\left(H\right)=0.9-0.75+0.5=0.65$
$P\left(H\right)=0.65$

Hence, Probability that the student will pass in Hindi is $0.65$