Question

Let X and Y be two events such that\[P\left( X \right){\text{ }} = \dfrac{1}{3}\], \[P\left( {X/Y} \right){\text{ }} = \dfrac{1}{2}\]and\[P{\text{ }}\left( {Y/X} \right){\text{ }} = \dfrac{2}{5}\].Then
(A). \[P\left( {X \cup Y} \right) = \dfrac{2}{5}\]
(B). \[P(Y) = \dfrac{4}{{15}}\]
(C). \[P\left( {X'/Y} \right) = \dfrac{1}{2}\]
(D). \[P\left( {X \cap Y} \right) = \dfrac{1}{5}\]

Answer 1

Hint: To solve the question, at first we have to apply simple probability formulae to estimate\[P(Y)\],\[P\left( {X \cap Y} \right)\],\[P\left( {X \cup Y} \right)\],\[P(Y)\], \[P(X')\]and\[P\left( {X'/Y} \right)\]respectively. Finally we will choose the correct option which matches the estimation value.

Complete step-by-step answer:
Given that the
\[P\left( X \right){\text{ }} = \dfrac{1}{3}\] ……………………………… (1)
\[P\left( {X/Y} \right){\text{ }} = \dfrac{1}{2}\] ……………………………… (2)
\[P{\text{ }}\left( {Y/X} \right){\text{ }} = \dfrac{2}{5}\] ……………………………….. (3)
We know the Baye’s formula which is given by,
\[
   \Rightarrow P\left( {X/Y} \right) = \dfrac{{P\left( {Y/X} \right)P(X)}}{{P(Y)}} \\
   \Rightarrow P(Y) = \dfrac{{P\left( {Y/X} \right)P(X)}}{{P\left( {X/Y} \right)}} \\
\] ………………………………………….. (4)
Substituting the values of eq. (1), (2) and (3) in eq. (4) we will get,
\[
  P(Y) = \dfrac{{P\left( {Y/X} \right)P(X)}}{{P\left( {X/Y} \right)}} \\
   = \dfrac{{\dfrac{2}{5} \times \dfrac{1}{3}}}{{\dfrac{1}{2}}} \\
   = \dfrac{4}{{15}} \\
\]……………………………………………. (5)
We know that, \[P\left( {X \cap Y} \right) = P(Y/X) \cdot P(Y)\] ………………… (6)
Substituting the values of eq. (1), and (3) in eq. (6) we will get,
\[P\left( {X \cap Y} \right) = \dfrac{2}{5} \times \dfrac{1}{3} = \dfrac{2}{{15}}\] ………………………………. (7)
Again we know that,
\[P\left( {X \cup Y} \right) = P(X) + P(Y) - P\left( {X \cap Y} \right)\] ………………………. (8)
Substituting the values of eq. (1), (5) and (7) in eq. (8) we will get,
\[
  P\left( {X \cup Y} \right) = \dfrac{1}{3} + \dfrac{4}{{15}} - \dfrac{2}{{15}} \\
   = \dfrac{{5 + 4 - 2}}{{15}} \\
   = \dfrac{7}{{45}} \\
\]……………………..……………………. (9)
Now the expression for \[P\left( {X'/Y} \right)\] is given by
\[P\left( {X'/Y} \right) = \dfrac{{P\left( {X' \cap Y} \right)}}{{P(Y)}} = \dfrac{{P(Y) - P(X \cap Y)}}{{P(Y)}}\] ………………………………………. (10)
Substituting the values of eq. (5) and (7) in eq. (12), we will get,
\[P\left( {X'/Y} \right) = \dfrac{{\dfrac{4}{{15}} - \dfrac{2}{{15}}}}{{\dfrac{4}{{15}}}} = \dfrac{1}{2}\] ……………………………………… (13)
Here we get among the options given only options (B) and (C) match with the estimated values.
Hence options (B) and (C) are correct.

Note: For determining the conditional probability Baye’s formula is useful. It states that probability of an event A is based on the sum of the conditional probabilities of event A given that event B has or has not occurred and the most important is that the events A and B must be independent, mathematically
\[p(B) = p\left( {B\left| A \right.} \right)p(A) + p\left( {B\left| {{A^c}} \right.} \right)p({A^c})\]