If P(AUB)=خ© can A and B be independent? [on hold]

let XX be a random variable with sample space Ω\Omega.If P(A∪B)=ΩP(A \cup B)=\Omega can AA and BB be independent?



1 Answer


We know P(A∪B)=P(A)+P(B)−P(A∩B)\mathsf P(A\cup B)=\mathsf P(A)+\mathsf P(B)-\mathsf P(A\cap B)

We are given that P(A∪B)=1\mathsf P(A\cup B)=1.

If the events A,BA,B are independent, then P(A∩B)=P(A)P(B)\mathsf P(A\cap B)=\mathsf P(A)\mathsf P(B).  

Thus if this is so, therefore: 1=P(A)+P(B)−P(A)P(B)1 = \mathsf P(A)+\mathsf P(B)-\mathsf P(A)\mathsf P(B)

Solve this equation for P(A)\mathsf P(A).



Did you see egreg’s comment?
– YoTengoUnLCD
Oct 20 at 23:42



Thank you @GrahamKemp! That is similar to what i thought of at first.I think the given information is deficient, not a very good question.
– Beth