The title says it all pretty much. What exactly (if anything) do conditional probabilities mean for independent events?
Example: I have two switches, A and B, in a wire, both of which have independent probabilities of working and I need both to work in order for the wire to work. Then it doesn't make sense if I try and calculate the probability of the wire working with P(A|B) or P(B|A) (because they would simply give me P(A) or P(B) ). I realize that the answer would be P(AB), but I it's interesting that I can't do it with conditional probability. It would seem then that there are no conditional probabilities for independent events.
Best Answer
It means exactly what you have said: $P(A\mid B)=P(A)$.
Moreover, this is in fact the intuitive idea behind independence: knowing that $B$ has occurred does not affect the probability of $A$ occurring.
Caveat: this does not hold if $P(B)=0$, because $P(A\mid B)$ is then undefined; whereas the standard definition of independence $P(A\cap B)=P(A)P(B)$ works fine if $P(B)=0$.