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?

Hint:
First, let $E$ be the event that a student passes in English and $H$ be the event that student passes in Hindi. After that, apply the condition as per the question and use De Morgan’s law. Try it, you will get the answer.

Complete step by step solution:
Let $E$ be the event that a student passes in English and $H$ be the event that student passes in Hindi.
It is given that the probability that a student will pass the final examination in both English and Hindi is $0.5$.
We can write it as,
$P(E\cap H)=0.5$ ………….. (1)
Also, the probability of passing neither is $0.1$.
$P({{E}^{'}}\cap {{H}^{'}})=0.1$
Now, the probability of passing the English examination is $0.75$ .
$P(E)=0.75$
Now, $P({{E}^{'}}\cap {{H}^{'}})=0.1$
Using De Morgan’s law we get,
De Morgan’s Law states that the complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements.
$({{A}^{'}}\cap {{B}^{'}})={{(A\cup B)}^{'}}$
$P({{E}^{'}}\cap {{H}^{'}})=P{{(E\cup H)}^{'}}=0.1$
We can write $P{{(E\cup H)}^{'}}=1-P(E\cup H)$
$P(E\cup H)=1-P{{(E\cup H)}^{'}}$
$P(E\cup H)=1-0.1$
Simplifying we get,
$P(E\cup H)=0.9$
Now we know that,
$P(E\cup H)=P(E)+P(H)-P(E\cap H)$
Now, putting values we get,
$0.9=0.75+P(H)-0.5$
Now simplifying we get,
$P(H)=0.65$

Therefore, the probability of passing the Hindi examination is $0.65$.

Additional information:
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen.

Note:
A well-defined collection of objects or elements is known as a set. Various operations like complement of a set, union and intersection can be performed on two sets. These operations and their usage can be further simplified using a set of laws known as De Morgan’s Laws.