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.
$P \left( X \mid Y _ { 1 } \right) = 0.2$ , $P \left( X \mid Y _ { 2 } \right) = 0.3$ , $P \left( X \mid Y _ { 3 } \right) = 0.6$ , $P \left( Y _ { 1 } \right) = 0.2$ , $P \left( Y _ { 2 } \right) = 0.4$ .

Find $P \left( Y _ { 1 } \mid X \right)$ .

A) $P \left( Y _ { 1 } \mid X \right) = 0.6$
B) $P \left( Y _ { 1 } \mid X \right) = 0.9$
C) $P \left( Y _ { 1 } \mid X \right) = 0.0072$
D) $P \left( Y _ { 1 } \mid X \right) = 0.3$
E) $P \left( Y _ { 1 } \mid X \right) = 0.1$

Question

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.
  $P \left( X \mid Y _ { 1 } \right) = 0.2$ , $P \left( X \mid Y _ { 2 } \right) = 0.3$ , $P \left( X \mid Y _ { 3 } \right) = 0.6$ , $P \left( Y _ { 1 } \right) = 0.2$ , $P \left( Y _ { 2 } \right) = 0.4$ .
 
Find $P \left( Y _ { 1 } \mid X \right)$ .
 
A) $P \left( Y _ { 1 } \mid X \right) = 0.6$
B)   $P \left( Y _ { 1 } \mid X \right) = 0.9$
C)   $P \left( Y _ { 1 } \mid X \right) = 0.0072$
D)   $P \left( Y _ { 1 } \mid X \right) = 0.3$
E)   $P \left( Y _ { 1 } \mid X \right) = 0.1$

Quizplus · Accepted Answer

Using Bayes' theorem: $P(Y_1 \mid X) = \frac{P(X \mid Y_1) \cdot P(Y_1)}{P(X)}$. First, calculate $P(X)$ using the law of total probability: $P(X) = P(X \mid Y_1)P(Y_1) + P(X \mid Y_2)P(Y_2) + P(X \mid Y_3)P(Y_3)$. Given $P(Y_3) = 0.4$ (since $P(Y_1) + P(Y_2) + P(Y_3) = 1$), $P(X) = 0.2 \cdot 0.2 + 0.3 \cdot 0.4 + 0.6 \cdot 0.4 = 0.32$. Then, $P(Y_1 \mid X) = \frac{0.2 \cdot 0.2}{0.32} = 0.1$.

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