Are these 3 events independent?

Probability theory and statistics

Are these 3 events independent?

Postby Guest » Mon Oct 07, 2019 4:34 am

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.
Guest
 

Re: Are these 3 events independent?

Postby Guest » Mon Oct 07, 2019 3:01 pm

Guest wrote: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.


[tex]P((A \cap B )\cup C)= P((A \cup B) \cap (B \cup C)) = P((A \cup B) \cup (B \cup C)) - P(A \cup B) - P(B \cup C)[/tex].

Hence, [tex]P((A \cap B )\cup C)= P((A \cup B) \cap (B \cup C)) = P(A \cup B) * P(B \cup C)[/tex] if sets [tex]A \cup B[/tex] and [tex]B \cup C[/tex] are independent.

Therefore, [tex]P(A \cap B \cap C)=P(A)*P(B)*P(C)[/tex] if only if the sets A, B, C are mutually independent.

Relevant Reference Links:

'Algebra of Sets',

https://en.wikipedia.org/wiki/Algebra_of_sets;

'Further Concepts in Probability',

https://www.wyzant.com/resources/lessons/math/statistics_and_probability/probability/further_concepts_in_probability.
Guest
 

Re: Are these 3 events independent?

Postby Guest » Tue Oct 08, 2019 10:10 am

Guest wrote:
Guest wrote: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.


[tex]P((A \cap B )\cup C)= P((A \cup B) \cap (B \cup C)) = P((A \cup B) \cup (B \cup C)) - P(A \cup B) - P(B \cup C)[/tex].

Hence, [tex]P((A \cap B )\cup C)= P((A \cup B) \cap (B \cup C)) = P(A \cup B) * P(B \cup C)[/tex] if sets [tex]A \cup B[/tex] and [tex]B \cup C[/tex] are independent.

Therefore, [tex]P(A \cap B \cap C)=P(A)*P(B)*P(C)[/tex] if only if the sets A, B, C are mutually independent.

Relevant Reference Links:

'Algebra of Sets',

https://en.wikipedia.org/wiki/Algebra_of_sets;

'Further Concepts in Probability',

https://www.wyzant.com/resources/lessons/math/statistics_and_probability/probability/further_concepts_in_probability.


A Minor Remark: [tex]P((A \cup B) \cup (B \cup C)) = P(A \cup B \cup C)[/tex].
Guest
 

Re: Are these 3 events independent?

Postby Guest » Tue Oct 08, 2019 11:29 pm

Hmm.

[tex]P((A \cap B )\cup C)= P((A \cup B) \cap (B \cup C)) = P(A \cup B) * P(B \cup C)[/tex] if sets, [tex]A \cup B[/tex] and [tex]B \cup C[/tex], are independent.

However, the sets, [tex]A \cup B[/tex] and [tex]B \cup C[/tex], are not independent. The set B is common to both of those sets even if A and B are mutually independent sets.

Therefore, [tex]P((A \cap B )\cup C)= P((A \cup B) \cap (B \cup C)) = P((A \cup B) \cup (B \cup C)) - P(A \cup B) - P(B \cup C)[/tex]
[tex]= P(A \cup B \cup C) - P(A \cup B) - P(B \cup C)[/tex].
Guest
 


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