# One way to check whether two events are independent is with the formula P(A&B) = P(A)*P(B). If this holds, the two events are independent. Now if A and B are mutually exclusive events, and P(A)>0 and P(B)>0, then P(A&B) = 0 ≠ P(A)*P(B), and thus the events are considered dependent.

One way to check whether two events are independent is with the formula $P\left(A\mathrm{&}B\right)=P\left(A\right)\ast P\left(B\right)$. If this holds, the two events are independent (to my knowledge).
Now if $A$ and $B$ are mutually exclusive events, and $P\left(A\right)>0$ and $P\left(B\right)>0$, then $P\left(A\mathrm{&}B\right)=0\ne P\left(A\right)\ast P\left(B\right)$, and thus the events are considered dependent. Why does this make intuitive sense?
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Bryant Liu
If two events are mutually exclusive, then one of them occurring forces the other not to occur. That is totally incompatible with independence, in which (knowledge of) one of them occurring does not affect the probability of the other occurring.
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muroscamsey
Mutually exclusive means "if one happens, the other doesn't". Which means that knowing something about whether one event happens gives you information about whether the other did.
Rolling a 3 on a die and flipping heads on a coin are independent events. They don't affect each other, you can't gain information about the result of one from knowing something about the result of the other. If I roll a 3 on the die and then ask you the probability that the coin flipped heads, you can safely ignore the die roll and tell me that it's 50%.
Flipping heads on a coin and flipping tails on the same toss are mutually exclusive (and, to some approximation, exhaustive) events. If I tell you the coin flipped tails, then ask you what the probability is that it flipped heads, you're going to give me a completely different answer to before.