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A. \[\dfrac{{27}}{{81}}\]

B. \[\dfrac{{12}}{{81}}\]

C. \[\dfrac{4}{{81}}\]

D. \[\dfrac{3}{{81}}\]

E. \[\dfrac{1}{{81}}\]

Answer

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Probability of choosing correct option for each of the 4 questions \[ = {\rm{p = }}{{\rm{p}}_{Q1}} = {{\rm{p}}_{Q2}} = {{\rm{p}}_{Q3}} = {{\rm{p}}_{Q4}} = \dfrac{1}{3}\].

Now to calculate the probability of choosing the correct option for all 4 questions, we use the rule of multiplication.

\[{\rm{p}}\left( {{{\rm{Q}}_1} \cap {{\rm{Q}}_2} \cap {{\rm{Q}}_3} \cap {{\rm{Q}}_4}} \right) = {\rm{p}}\left( {{{\rm{Q}}_1}} \right){\rm{p}}\left( {{{\rm{Q}}_2}} \right){\rm{p}}\left( {{{\rm{Q}}_3}} \right){\rm{p}}\left( {{{\rm{Q}}_4}} \right)\] [since all event are independent]

\[ = \dfrac{1}{3} \times \dfrac{1}{3} \times \dfrac{1}{3} \times \dfrac{1}{3} = \dfrac{1}{{81}}\]

Thus the required probability is \[\left( {\rm{D}} \right)\dfrac{1}{{81}}\]