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Mutually Exclusive: Occurrence of one event means the other event can not occur.
For e.g. occurrence of Tail and Head on flipping a coin are Mutually Exclusive events. If 'H' occurs 'T' can't occur and vice versa. In case of M.E events P[A+C]= P[A]+P[C]
Mutually Independent : Occurrence of one event does not affect the occurrence other event.
For e.g. When flipping two coins, getting 'T' on both the coins is Mutually Independent. 'T' occurring on first coin has no bearing on 'T' on other coin and 'T' can occur in both the coins (getting 'T' on first coin does not mean we can not get 'T' on second coin hence they are not mutually exclusive events)
For M.I events we have : P[A+C] = P[A] x P[C]
I hope it helps. Happy to be corrected if anyone can.
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