Let A, B, C be three mutually independent events. Consider the two statements S₁ and S₂.
S₁ : A and B $\cup$ C are independent.
S₂ : A and B $\cap$ C are independent.
Then

Question

Let A, B, C be three mutually independent events. Consider the two statements $$S1$$ and $$S2$$.$$S1$$ : A and B ∪ C are independent.$$S2$$ : A and B ∩ C are independent.Then

Infinity Learn · Accepted Answer

We are given that                      $$P(A$$∩$$B)=P(A)P(B)P(B$$∩$$C)=P(B)P(C)P(C$$∩$$A)=P(C)P(A)P(A$$∩B∩$$C)=P(A)(B)P(C)We$$ have,              $$P(A$$∩(B$$∩$$C))=P(A$$∩B∩$$C)=P(A)P(B)P(C)=P(A)P(B$$∩$$C)⇒ A and B a C are independentTherefore, $$S2$$ is true. Also,                  P[(A$$∩$$(B$$∪$$C)]$$=P$$[(A$$∩$$B)∪(A$$∩$$C)]  $$=P(A$$∩$$B)+P(A$$∩$$C)$$−P[(A$$∩$$B)∩(A$$∩$$C)]  $$=P(A$$∩$$B)+P(A$$∩$$C)$$−$$P(A$$∩B∩$$C)$$  $$=P(A)P(B)+P(A)P(C)$$−$$P(A)P(B)P(C)$$  $$=P(A)$$[$$P(B)+P(C)$$−$$P(B)P(C)$$]  $$=P(A)$$[$$P(B)+P(C)$$−$$P(B$$∩$$C)$$]  $$=P(A)P(B$$∪$$C)Therefore$$, A and B ∪ C are independent

Let A, B, C be three mutually independent events. Consider the two statements S₁ and S₂.
S₁ : A and B $\cup$ C are independent.
S₂ : A and B $\cap$ C are independent.
Then

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Let A, B, C be three mutually independent events. Consider the two statements S1 and S2.S1 : A and B ∪ C are independent.S2 : A and B ∩ C are independent.Then

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Let A, B, C be three mutually independent events. Consider the two statements S₁ and S₂.
S₁ : A and B $\cup$ C are independent.
S₂ : A and B $\cap$ C are independent.
Then