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De Morgan's Law

There are a couple of equivalent set notations on probabilities and they are called De Morgan’s Laws.

(A∩B)′=(A′∪B′) for sets or P(A∩B)′=P(A′∪B′) for probabilities.

and

(A∪B)′=(A′∩B′) for sets or P(A∪B)′=P(A′∩B′) for probabilities.

Now, let's solve the following problems.

  1. Given P(A)=0.6P(B)=0.3 and P(A∪B)=0.7, find P(A∩B) and P(A′∪B′).

First, we can use the formula for the union of two sets to determine the intersection.

P(A)+P(B)−P(A∩B)0.6+0.3−P(A∩B)0.9−0.70.2=P(A∪B)=0.7=P(A∩B)=P(A∩B)

Now we can use De Morgan’s Law to find P(A′∪B′).

 

P(AB)=P(AB)=1P(AB)=10.2=0.8.

P(AB)=P(AB)=1P(AB)=10.2=0.8.

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