Question

events ( a_1 ) and ( a_2 ) are mutually exclusive and form a complete partition of a sample space ( s ) with ( p(a_2)=0.37 ). if ( e ) is an event in ( s ) with ( p(e|a_1)=0.17 ) and ( p(e|a_2)=0.09 ), compute ( p(a_2|e)=? ) (hint: because ( a_1 ) and ( a_2 ) are mutually exclusive and form a complete partition of the sample space, ( p(a_1)=1 - p(a_2) )).
note: if your final answer has up to four decimal places, please enter your full answer without rounding it. if your answer contains more than four decimal places, please round it to four decimal places before entering it in the box below.

Explanation:

Step1: Find \( P(A_1) \)

Since \( A_1 \) and \( A_2 \) form a complete partition, \( P(A_1)=1 - P(A_2) \). Given \( P(A_2) = 0.37 \), then \( P(A_1)=1 - 0.37=0.63 \).

Step2: Use the law of total - probability to find \( P(E) \)

The law of total - probability states that \( P(E)=P(E|A_1)P(A_1)+P(E|A_2)P(A_2) \).
Substitute \( P(E|A_1) = 0.17 \), \( P(A_1)=0.63 \), \( P(E|A_2)=0.09 \), and \( P(A_2) = 0.37 \) into the formula:
\( P(E)=(0.17\times0.63)+(0.09\times0.37)=0.1071 + 0.0333=0.1404 \).

Step3: Use Bayes' theorem to find \( P(A_2|E) \)

Bayes' theorem is \( P(A_2|E)=\frac{P(E|A_2)P(A_2)}{P(E)} \).
Substitute \( P(E|A_2)=0.09 \), \( P(A_2) = 0.37 \), and \( P(E)=0.1404 \) into the formula:
\( P(A_2|E)=\frac{0.09\times0.37}{0.1404}=\frac{0.0333}{0.1404}\approx0.2372 \).

Answer:

\( 0.2372 \)