In probability is there a difference between sets that are disjoint and sets that are independent.
I don't know what it means for sets to be independent! I think you are asking about events that are mutually exclusive or independent. Two events, as sets of possible outcomes, are disjoint if and only if they the events are mutually exclusive. but you can't use the word "independent" in that way.
Yes, there is a very large difference! If two events are independent that means knowing one happens doesn't affect the probability that the other happens. If two events are mutually exclusive know that one happens means the probability of the other is 0! That certainly affects their probability.
If I roll a single die, what is the probability it comes up "5"? If I tell you the result was an even number what does that tell you about the probability it was a "5"?
In the measure-theoretic axiomatization of probability, you can regards sets as events, so this question is somewhat well-formed. And yes, there is a difference. You would say events (sets) A and B are independent if P(A n B)=P(A)P(B). If A and B are disjoint, then A n B = emptyset, so P( A n B ) = 0, so A and B are independent iff P(A)=0 or P(B)=0.
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