Independence of events and intersection

Asked May 08 '21 at 12:50

Active May 08 '21 at 14:20

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Suppose an event $A$ is independent of event $B$. Event $A$ is also independent of event $C$. We also know that $B$ and $C$ are dependent.

Is it true that $A$ is independent of $B\cap C$? I think the answer is yes since if the presence of $A$ doesn't affect either $B$ or $C$, it won't affect their intersection as well. But how do we prove it mathematically?

edited May 08 '21 at 14:20

asked May 08 '21 at 12:50

Patrick 88

Suppose $A$ is the event "a fair penny comes up $H$", suppose $B$ is the event "a fair dime comes up $H$", suppose $C$ is the event "the fair penny and the fair dime match." – lulu May 08 '21 at 12:52
5

Does this answer your question? If $A$ is independent of $B$ and $C$ then why is it not necessarally independent of $B\cap C$? – HappyChappy28 May 08 '21 at 12:55

Independence of events and intersection

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