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.
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?

Solution:

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., $P\left(A^{\prime} \cap B^{\prime}\right)=0.1$

P(A) = 0.75

Now, $(A \cup B)^{\prime}=\left(A^{\prime} \cap B^{\prime}\right) \quad[$ De Morgan’s law $]$

$\therefore \mathrm{P}(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=0.1$

$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})^{\prime}=1-0.1=0.9$

We know that P(A or B) = P(A) + P(B) – P(A and B)

$\therefore 0.9=0.75+\mathrm{P}(\mathrm{B})-0.5$

$\Rightarrow P(B)=0.9-0.75+0.5$

$\Rightarrow P(B)=0.65$

Thus, the probability of passing the Hindi examination is 0.65.

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