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October 7th, 2019, 12:33 AM   #1
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Are these events independent?

Hello everyone.

Let us consider 3 events A,B,C such that:

$$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$

Notice that the second term is a union and not an intersection.

Are they independent?

And what if the assumption was: $$P(A \cap( B \cup C))=P(A)*P(B)*P(C)$$?

I know that the independence condition requires us to check whether the probability of the intersection of each pair factorizes plus the probability of the intersection of all of them factorizes as well.

But I do not know how to prove that they are/they are not independent

Thank you.
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October 7th, 2019, 07:30 AM   #2
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Originally Posted by Patt View Post
Hello everyone.
$$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$
$$P((A \cap B )\cup C) = [P(A) \times P(B)] + P(C) - [P(A) \times P(B) \times P(C)]$$
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