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A sample space consists of 9 elementary ...

A sample space consists of 9 elementary outcomes `E_(1),E_(2),…..,E_(9)` whose probabilities are
`P(E_(1))=P(E_(2))=0.08,P(E_(3))=P(E_(4))=P(E_(5))=0.1`
`P(E_(6))=P(E_(1))=0.2,P(E_(8))=P(E_(9))=0.07`
`"Suppose"" "A={E_(1),E_(5),E_(8)},B={E_(2),E_(5),E_(8),E_(9)}`
(i) Calculate P(A), P(B) and `P(AcapB)`.
(ii) Using the addition law of probability, calculate `P (AcupB).`
(iii) List the composition of the event `AcupB` and calculate `P(AcupB)` by adding the probabilities of the elementary outcomes.
Calculate `P(overset(-)B)` from P(B), also calculate `P(overset(-)B)` directly from the elementary outcomes of `overset(-)B`,

Text Solution

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`"Given "S={E_(1),E_(2),E_(3),E_(4),E_(5),E_(6),E_(7),E_(8),E_(9)}`
`A={E_(1),E_(5),E_(8)},B={E_(2),E_(5),E_(8),E_(9)}`
`P(E_(1))=P(E_(2))=0.08`
`P(E_(3))=P(E_(4))=P(E_(5))=0.1`
`P(E_(6))=P(E_(7))=2,P(E_(8))=P(E_(9))=0.07`
(i) `P(A)=P(E_(1))+P(E_(5))+P(E_(8))`
`=0.08+0.1+0.07=0.25`
(ii) `P(AcupB)=P(A)+P(B)-P(AcapB)`
`"Now, "P(B)=P(E_(2))+P(E_(5))+P(E_(8))+P(E_(9))`
`=0.08+0.1+0.07+0.07=0.32`
`AcapB={E_(5),E_(8)}`
`P(AcapB)=P(E_(5))+P(E_(8))=0.1+0.7=0.17`
`"On substituting these values in Eq. (i), we get"`
`P(AcupB)=0.25+0.32-0.17=0.40`
(iii) `AcupB={E_(1),E_(2),E_(5),E_(8),E_(9)}`
`P(AcupB)=P(E_(1))+P(E_(2))+P(E_(5))+P(E_(8))+P(E_(9))`
`=0.08+0.08+0.1+0.07+0.07=0.40`
(iv) `becausePoverset(-)((B))=1-P(B)=1-0.32=0.68`
`"and "overset(-)B={E_(1),E_(3),E_(4),E_(6),E_(7)}`
(iv) `because" "Poverset(-)((B))=P(E_(1))+P(E_(3))+P(E_(4))+P(E_(6))+P(E_(7))`
`=0.08+0.1+0.1+0.2+0.2=0.68`
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