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Addition Theorem of Probability 

• Two NonDisjoint (NonMutually Exclusive) Events
For two events "A" and "B" which are not disjoint (or not mutually exclusive), the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is given by
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) • Three NonDisjoint (NonMutually Exclusive) Events
For three events "A", "B" and "C" which are not disjoint (or not mutually exclusive), the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is given by
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C) 
Addition Theorem of Probability » Mutually Exclusive Events 

• Mutually Exclusive
Two or more events are said to be mutually exclusive if the occurrence of one prevents the occurrence of the others. This can also be interpreted as the events not having any element in common.
» Two Events
Two events are mutually exclusive only if they do not have an element in common.
» Three Events
Three events are mutually exclusive only if they do not have an element in common.
• Pair wise Exclusive : Mutually Exclusive
Where three or more events are in consideration, the events would be mutually exclusive if and only if they are pairwise mutually exclusive.
Mutually Exclusive ⇔ Pair wise Exclusive. • Intersection of Mutually Exclusive Events = Φ
Since there would be no common elements in case of mutually exclusive events, the event representing the intersection of the events is an impossible event. The probability of occurrence of the intersection of mutually exclusive events is Nil.
Where "A" and "B" are mutually exclusive events, A ∩ B = {} ⇒ n (A ∩ B) = 0. Where "P", "Q" and "R" are mutually exclusive events, P ∩ Q ∩ R = Φ ⇒ n (P ∩ Q ∩ R) = 0.
• Two Disjoint (Mutually Exclusive) Events
For two events "A" and "B" which are disjoint (or mutually exclusive), the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is given by the sum of the probabilities of the individual events.
P(A ∪ B) = P(A) + P(B) • Three Disjoint (Mutually Exclusive) Events
For three events "A", "B" and "C" which are disjoint (or mutually exclusive), the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is given by the sum of the probabilities of the individual events.
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) 
Addition Theorem of Probability » Exhaustive Events 

Two or more events are said to be mutually exclusive if they cover between them all possible elementary events in relation to the experiment. The union set of exhaustive events is the set of all elementary events in relation to the experiment i.e. its sample space.
Where
Events "E' and "O" together form exhaustive events, since E ∪ O = S Any events combined with the exhaustive events ("E" and "O" here) would also form exhaustive events,
Probability of occurrence of the sample space is a certainty i.e. its probability is 1.
Since the union of exhaustive events is equal to the sample space, the probability of occurrence of the event representing the union of exhaustive events is a certainty i.e. its probability is 1. Where E ∪ O = S, P(E ∪ O) = P(S) ⇒ P(E ∪ O) = 1 • Two Exhaustive Events
For two events "A" and "B" which are exhaustive, the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty.
P(A ∪ B) = P(S) = 1 • Three Exhaustive Events
For three events "A", "B" and "C" which are exhaustive, the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty.
P(A ∪ B ∪ C) = P(S) = 1 
Addition Theorem of Probability » Mutually Exclusive and Exhaustive Events 

• Two Events
For two events "A" and "B" which are mutually exclusive and exhaustive, the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty and is given by the sum of the probabilities of the individual events.
• Three Exhaustive Events
For three events "A", "B" and "C" which are mutually exclusive and exhaustive, the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty and is given by the sum of the probabilities of the individual events.

Probability » Additional relations from set theory 

• Two Events
For any two events "A" and "B"
• Three Events
For any three events "A", "B" and "C"

Occurrence of Atleast, Atmost, Only, None of two or more events 

• Atleast
For any two or more events occurrence of atleast one of the events implies the occurrence of the event representing the union of those events
• Atmost
For any two events occurrence of atmost one of the events implies the non occurrence of the event representing the intersection of those events.
For any three events occurrence of atmost two of the events implies the non occurrence of the event representing the intersection of those events.
For any three events occurrence of atmost one of the events implies the non occurrence of the events representing the intersections of those events.
• Only
The use of the word only indicates the condition where only the mentioned event should occur and all other events should not occur.
For two events "A" and "B" occurrence of only one of the events
For three events "P", "Q", and "R" occurrence of only one of the events
For three events "P", "Q", and "R" occurrence of only two of the events

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