I know how to prove two events to be independent by proving $P(E|F) = P(E)$ but is there any trick or a underlying concept that I never knew of. I feel like I am missing something here.
2026-04-01 00:11:04.1775002264
How to determine independency of two events just by looking at them?
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In many situations independence can (and should) be deduced from the context.
Imagine we have a deck of 53 cards, extract a card randomly (call it $C_1$), return it to the deck and extract a second card (call it $C_2$.) The events $C_1$ is an ace and $C_2$ is a ace are independent. The result of the first extraction does not affect the probabilities of the second, because we have returned the first card to the deck.
Now we do as before but we do not return the first card to the deck. Then the events $C_1$ is an ace and $C_2$ is a ace are not independent, since the result of the first extraction affects the probabilities of the second.