Correct Answer - Option 1 : 1 and 0.5
Concept:
Sample space: The collection of all possible outcomes of the random experiment is known as a sample space
Event: The outcome of an experiment is known as event
Probability: The probability of an event is defined as the ratio of favorable cases of an event to the total number of ways possible. It deals with the occurrence of random variable.
If the events are independent then P(X ∩ Y) = P(X) × P(Y)
\({\bf{P}}\left( {\frac{{\bf{X}}}{{\bf{Y}}}} \right)\) = \(\frac{{{\bf{P}}\left( {{\bf{X}} \cap {\bf{Y}}} \right)}}{{{\bf{P}}\left( {\bf{Y}} \right)}}\)
Calculation:
Given:
P(X) = 1 and P(Y) = 0.5
(i) \({\bf{P}}\left( {\frac{{\bf{X}}}{{\bf{Y}}}} \right)\)
\({\bf{P}}\left( {\frac{{\bf{X}}}{{\bf{Y}}}} \right)\) = \(\frac{{{\bf{P}}\left( {{\bf{X}} \cap {\bf{Y}}} \right)}}{{{\bf{P}}\left( {\bf{Y}} \right)}}\)
\(\frac{{{\rm{P}}\left( {{\rm{X}} \cap {\rm{Y}}} \right)}}{{{\rm{P}}\left( {\rm{Y}} \right)}} = {\rm{\;}}\frac{{{\rm{P}}\left( {\rm{X}} \right) \times {\rm{P}}\left( {\rm{Y}} \right)}}{{{\rm{P}}\left( {\rm{Y}} \right)}} = {\rm{\;}}\frac{{1 \times 0.5}}{{0.5}} = 1\)
∴ \({\bf{P}}\left( {\frac{{\bf{X}}}{{\bf{Y}}}} \right)\) = 1
(ii) \({\bf{P}}\left( {\frac{{\bf{Y}}}{{\bf{X}}}} \right)\)
\({\rm{P}}\left( {\frac{{\rm{Y}}}{{\rm{X}}}} \right)\) = \(\frac{{{\rm{P}}\left( {{\rm{Y}} \cap {\rm{X}}} \right)}}{{{\rm{P}}\left( {\rm{X}} \right)}} = {\rm{\;}}\frac{{{\rm{P}}\left( {\rm{Y}} \right) \times {\rm{P}}\left( {\rm{X}} \right)}}{{{\rm{P}}\left( {\rm{X}} \right)}} = {\rm{\;}}\frac{{0.5 \times 1}}{1} = 0.5\)
∴ \({\bf{P}}\left( {\frac{{\bf{Y}}}{{\bf{X}}}} \right)\) = 0.5
