Prove that if E and F are independent events, then the events E and F�

1.	Prove that if E and F are independent events, then the events E and F′ are also independent.
Answer» Since, E and F are independent events. ⇒ P(E ∩ F) = P(E). P(F) Now, P(E ∩ F') = P(E). P(E ∩ F) = P(E) – P(E). P(F) = P(E)(1 – P(F)) ⇒ P(E ∩ F′) = P(E). P(F′) Hence, E and F′ are independent events.

Prove that if E and F are independent events, then the events E and F′ are also independent.

Answer»

Since, E and F are independent events.

⇒ P(E ∩ F) = P(E). P(F)

Now, P(E ∩ F') = P(E). P(E ∩ F)

= P(E) – P(E). P(F) = P(E)(1 – P(F))

⇒ P(E ∩ F′) = P(E). P(F′)

Hence, E and F′ are independent events.

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Prove that if E and F are independent events, then the events E and F′ are also independent.

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