How to prove that:?

Question

A and B and C are mutually independent events, so A and BuC are independent

Answer 1

Don't believe the other answerers: the statement that A,B and C are mutually independent does not mean that AnB=null etc., that would be mutually exclusive!

Mutual independence means that
P(AnB)=P(A)*P(B),
P(AnC) = P(A)*P(C),
P(BnC) = P(B)*P(C), and
P(AnBnC) = P(A)*P(B)*P(C).

Assuming this, you need to show that P(An(BuC)) = P(A)*P(BUC) (Eq. I.).

First write An(BuC) = (AnB)u(AnC) by the distributive law.
Then apply the inclusion and exclusion principle, which says that P(AuB) = P(A)+P(B)-P(AnB), to the left hand side of Eq. I..

This gives P(An(BuC)) = P((AnB)u(AnC)) = P(AnB) + P(AnC) - P((AnB)n(AnC)) = P(A)*P(B) + P(A)*P(C) - P(AnBnC) = P(A)*P(B) + P(A)*P(C) - P(A)*P(B)*P(C).

Applying the inclusion and exclusion principle to the right-hand side of Eq. I. gives P(A)*P(BUC) = P(A)*(P(B)+P(C)-P(BnC)) = P(A)*P(B) + P(A)*P(C) - P(A)*P(BnC) = P(A)*P(B) + P(A)*P(C) - P(A)*P(B)*P(C).

This implies that the desired result holds.

Answer 2

I get it you're using u as Union, so we'll also use n as Intersection
Since, A, B, C are mutually independent then:
A n B = 0
A n C = 0

now A n (BuC) = (AnB) u (AnC) = 0 u 0 = 0. So there is not intersection between A & BuC, thus they are mutually independent. QED.

Answer 3

A intersection BuC
= (Aintersection B) U (Aintersection C)
= empty U empty
=empty

Answer 4

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Answer 5

Mutually independent means that
A∩B = null
A∩C = null
B∩C = null

A∩(BuC) = (A∩B)u(A∩C) = (null)u(null) = null

Answer 6

If there is no connection of any kind between them, then they are independent.

Answer 7

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