Use Bayes' Theorem or a Tree Diagram to Calculate the Indicated

Multiple Choice

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places. $Y _ { 1 }$ , $Y _ { 2 }$ , $Y _ { 3 }$ form a partition of S.
$P \left( X \mid Y _ { 1 } \right) = 0.2$ , $P \left( X \mid Y _ { 2 } \right) = 0.3$ , $P \left( X \mid Y _ { 3 } \right) = 0.5$ , $P \left( Y _ { 1 } \right) = 0.3$ , $P \left( Y _ { 2 } \right) = 0.4$ .

Find $P \left( Y _ { 2 } \mid X \right)$ .

A) $P \left( Y _ { 2 } \mid X \right) = 0.009$
B) $P \left( Y _ { 2 } \mid X \right) = 0.1818$
C) $P \left( Y _ { 2 } \mid X \right) = 0.6364$
D) $P \left( Y _ { 2 } \mid X \right) = 0.3636$
E) $P \left( Y _ { 2 } \mid X \right) = 0.4545$