whats the basic difference between mutually exclusive events and independent events ? how do both associate with each other ? and can both of them occur simultaneously ?


Totally Different Definitions.

Mutually Exclusive: Both cannot occur simultaneously, i.e. $A\cap B=\emptyset.$

Independent: $P(A\cap B)=P(A) P(B),$ or in terms of conditional probabilities $P(B|A)=P(B)$ and $P(A|B)=P(A).$ In other words, the outcome of $A$ does not affect the probability $B$ will occur, and vice versa.


Mutually exclusive events $A$ and $B$ are events that cannot both happen (at the same time, for the same situation), for example it is not possible to get both heads and tails on a single flip of a coin. Mathematically:

$$P(A\cap B)=0$$ and so $$P(A\cup B)=P(A)+P(B)$$ for mutually exclusive events (no intersection on a Venn diagram).

Independent events $C$ and $D$ are events that don't affect each other, for example getting a 6 on a die and it raining tomorrow (presumably!). Mathematically:

$$P(C|D)=P(C)$$ $$P(D|C)=P(D)$$ or $$P(C\cap D) = P(C)\times P(D)$$


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