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# The probability that a student will pass the final examination in both English and Hindi 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?

$\begin{array}{1 1}(A)\;0.65\\(B)\;0.75\\(C)\;0.85\\(D)\;0.35\end{array}$

Toolbox:
• $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
• $P(A)+P(A')=1$
Step 1:
Let H denote Hindi,E denote English
Given :
$P(H\cap E)=0.5$
P(passing neither)=$P(H'\cap E')=P(H \cup E)'=0.1$
$\therefore P(H \cup E)=1-P(H\cup E)'$
$\Rightarrow 1-0.1$
$\Rightarrow 0.9$
Step 2:
Also given $P(E)=0.75$
$\therefore P(H\cup E)=P(H)+P(E)-P(H\cap E)$
$0.9=P(H)+0.75-0.5$
$\;\;\;\;\;\;=P(H)+0.25$
$0.9-0.25=P(H)$
$\Rightarrow P(H)=0.65$
$\therefore$ Probability of passing Hindi exam is 0.65
Hence (A) is the correct answer.