Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places. $P ( A \mid B ) = 0.8$ , $P ( B ) = 0.2$ , $P \left( A \mid B ^ { \prime } \right) = 0.3$ . Find $P ( B \mid A )$ .

A) $P ( B \mid A ) = 0.16$
B) $P ( B \mid A ) = 0.12$
C) $P ( B \mid A ) = 0.4$
D) $P ( B \mid A ) = 0.6$
E) $P ( B \mid A ) = 0.84$

Question

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places.   $P ( A \mid B ) = 0.8$ , $P ( B ) = 0.2$ , $P \left( A \mid B ^ { \prime } \right) = 0.3$ . Find $P ( B \mid A )$ .
 
A) $P ( B \mid A ) = 0.16$
B)   $P ( B \mid A ) = 0.12$
C)   $P ( B \mid A ) = 0.4$
D)   $P ( B \mid A ) = 0.6$
E)   $P ( B \mid A ) = 0.84$

Quizplus · Accepted Answer

To find $P(B|A)$, we use Bayes' theorem: $P(B|A) = \frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B')P(B')}$. Given $P(A|B) = 0.8$, $P(B) = 0.2$, and $P(A|B') = 0.3$, with $B'$ being the complement of $B$, thus $P(B') = 1 - P(B) = 0.8$. Substituting these values in, we get $P(B|A) = \frac{0.8 \times 0.2}{0.8 \times 0.2 + 0.3 \times 0.8} = \frac{0.16}{0.16 + 0.24} = \frac{0.16}{0.4} = 0.4$.

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