Question

A sample space consists of 9 elementary event E1,E2,E3.....E8,E9 whose probabilities are P(E1)=P(E2)=0.08 ,P(E3)=P(E4)=0.1 , P(E6)=P(E7)=0.2 ,P(E8)=P(E9)=0.07 . Suppose A={E1,E5,E8} , B={E2,E5,E8,E9} . Compute P(A) , P(B) and P(A∩B) . Using the addition law of probability, find P(A∪B) . List the composition of the event A∪B , and calculate, P(A∪B) by adding the probabilities of the elementary events. Calculate P( ˉ B ) from P(B) , also calculate P( ˉ B ) directly from the elementary events of ˉ B . Select the correct answer from above options

Answer 1

Here, P(E1)=P(E2)=0.08 P(E3)=P(E4)=0.1 P(E6)=P(E7)=0.2 P(E8)=P(E9)=0.07 First we will calculate P(E5). As sample space contains events from E1 to E9. ∴P(E1)+P(E2)+P(E3)+P(E4)+P(E5)+P(E6)+P(E7)+P(E8)+P(E9)=1 ∴ P ( E 1 ) + P ( E 2 ) + P ( E 3 ) + P ( E 4 ) + P ( E 5 ) + P ( E 6 ) + P ( E 7 ) + P ( E 8 ) + P ( E 9 ) = 1 ⇒0.08+0.08+0.1+0.1+P(E5)+0.2+0.2+0.07+0.07=1 ⇒0.9+P(E5)=1 ⇒P(E5)=0.1 Now, P(A)=P(E1)+P(E5)+P(E8)=0.08+0.1+0.07=0.25 P(B)=P(E2)+P(E5)+P(E8)+P(E9)=0.08+0.1+0.07+0.07=0.32 P ( B ) = P ( E 2 ) + P ( E 5 ) + P ( E 8 ) + P ( E 9 ) = 0.08 + 0.1 + 0.07 + 0.07 = 0.32 P(A∩B)=P(E5)+P(E8)=0.1+0.07=0.17 ∴P(A∪B)=P(A)+P(B)-P(A∩B)=0.25+0.32-0.17=0.4 Now, A∪B={E1,E2,E5,E8,E9} ∴P(A∪B)=P(E1)+P(E2)+P(E5)+p(E8)+P(E9)=0.08+0.08+0.1+0.07+0.07=0.4 Now, P( ˉ B )=1-P(B)=1-0.32=0.68 Also, ˉ B ={E1,E3,E4,E6,E7} ∴P( ˉ B )=P(E1)+P(E3)+P(E4)+P(E6)+P(E7)=0.08+0.1+0.1+0.2+0.2=0.68