A and B of a sample space S, but assume that Pr[A]=0.2 and Pr[B]=0.6. Find Pr[A∪B]...
A and B of a sample space S, but assume that Pr[A]=0.2 and Pr[B]=0.6. Find Pr[A∪B] under each of the following conditions: (1) If A⊂B, then Pr[A∪B]= (2) If A∩B=∅, then Pr[A∪B]= (3) If A∩B′=∅, then Pr[A∪B]=
Homework Answers
Given that, Pr[A] = 0.2 and Pr[B] = 0.6
We want to find, Pr[A∪B] under each of the following conditions.
1) If A⊂B, then Pr[A ∩ B] = Pr[A]
Pr[A∪B]
= Pr[A] + Pr[B] - Pr[A ∩ B]
= Pr[A] + Pr[B] - Pr[A]
= 0.2 + 0.6 - 0.2
= 0.6
=> Pr[A∪B] = 0.6
2) If A∩B=∅, then Pr[A∩B] = 0
Pr[A∪B] = Pr[A] + Pr[B] - Pr[A∩B] = 0.2 + 0.6 - 0
=> Pr[A∪B] = 0.8
3) If A∩B′=∅, then
Pr[A ∩ B'] = Pr[A] - Pr[A∩B]
=> 0 = Pr[A] - Pr[A ∩ B]
=> Pr[A ∩ B] = Pr[A]
Therefore,
Pr[A∪B]
= Pr[A] + Pr[B] - Pr[A ∩ B]
= Pr[A] + Pr[B] - Pr[A]
= 0.2 + 0.6 - 0.2
= 0.6
=> Pr[A∪B] = 0.6
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