Question

question 6
5 pts
events ( a_1 ) and ( a_2 ) are mutually exclusive and form a complete partition of a sample space ( s ) with ( p(a_1)=0.65 ). if ( e ) is an event in ( s ) with ( p(e|a_1)=0.18 ) and ( p(e|a_2)=0.11 ), compute ( p(e)=? ) (hint: because ( a_1 ) and ( a_2 ) are mutually exclusive and form a complete partition of the sample space, ( p(a_2)=1 - p(a_1) )).
note: if your final answer has up to four decimal places, pleas 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: Calculate \(P(A_2)\)

Since \(A_1\) and \(A_2\) are mutually exclusive and form a complete partition of the sample space, \(P(A_2)=1 - P(A_1)\). Given \(P(A_1) = 0.65\), then \(P(A_2)=1 - 0.65=0.35\).

Step2: Use the law of total probability

The law of total probability states that \(P(E)=P(E|A_1)P(A_1)+P(E|A_2)P(A_2)\).
Substitute the known values: \(P(E|A_1) = 0.18\), \(P(A_1)=0.65\), \(P(E|A_2)=0.11\), and \(P(A_2) = 0.35\).

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Answer:

\(0.1555\)