# Probability Addition Theorem Probability of Atmost, Atleast, Neither, All One or More Events

## Addition Theorem of Probability - Non Mutually Exclusive (Non Disjoint) Events

### Two Non-Disjoint (Non-Mutually 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 Non-Disjoint (Non-Mutually 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 (Disjoint) Events

### Intersection of Mutually Exclusive Events = Φ

#### Mutually Exclusive

Two or more events are said to be mutually exclusive if the occurrence of one prevents the occurrence of the others.

Three or more events taken together would form mutually exclusive events if and only if they are pairwise mutually exclusive.

Mutually exclusive property can be interpreted as the events not having any element in common i.e. their intersection being a null set.

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 = {Φ} ⇒ n (A ∩ B) = 0.

Thus P(A ∩ B) =
 n(A ∩ B) n(S)
=
 0 n(S)
= 0

Where "P", "Q" and "R" are mutually exclusive events,

P ∩ Q ∩ R = {Φ} ⇒ ⇒ n (P ∩ Q ∩ R) = 0.

Thus P(P ∩ Q ∩ R) =
 n(P ∩ Q ∩ R) n(S)
=
 0 n(S)
= 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 is given by the sum of the probabilities of the individual events

Probability of the occurrence of the union of the events

 = 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 is given by the sum of the probabilities of the individual events.

Probability of the occurrence of the union of the events

 = sum of the probabilities of the individual events. ⇒ P(A ∪ B ∪ C) = P(A) + P(B) + P(C)

## Addition Theorem of Probability - Exhaustive Events

Occurrence of one of the events from the sample space is a certainity.

⇒ Probability of occurrence of the sample space is 1.
⇒ P(S) = 1

### Probability of Union of Exhaustive Events

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.

 P(A ∪ B) = P(A) + P(B) [Mutually Exclusive] [Exhaustive] [Mutually Exclusive and Exhaustive]

### 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.

 P(A ∪ B ∪ C) = P(A) + P(B) + P(C) [Mutually Exclusive] [Exhaustive] [Mutually Exclusive and Exhaustive]

## Probability - Relations derived from Set theory

### Two Events

For any two events "A" and "B"
• P(A) = P(A ∩ B) + P(A ∩ Bc)
• P(B) = P(A ∩ B) + P(Ac ∩ B)
• P(A ∪ B)c = P(Ac ∩ Bc)
• P(A ∩ B)c = P(Ac ∪ Bc)

### Three Events

For any three events "A", "B" and "C"
• P(A ∪ B ∪ C)c = P(Ac ∩ Bc ∩ Cc)
• P(A ∩ B ∩ C)c = P(Ac ∪ Bc ∪ Cc)

## Probability of Occurrence of atleast one or more of two or more events

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

### Atleast one of the two events

• At least one of the events "A" and "B" ⇒ The event A ∪ B  P(A ∪ B) = 1 − P(A ∪ B)c = 1 − P(Ac ∩ Bc)

At least one of the events is a complimentary of none [P(Ac ∩ Bc)] of the events.

### Atleast one of the three events

• At least one of the events "P", "Q" and "R" ⇒ The event P ∪ Q ∪ R  P(P ∪ Q ∪ R) = 1 − P(P ∪ Q ∪ R)c = 1 − P(Pc ∩ Qc ∩ Rc)

At least one of the events is a complimentary of none [P(Pc ∩ Qc ∩ Rc)] of the events.

### Atleast two of the three events

• At least two of the events "P", "Q" and "R"

⇒ The union of the events

• P(Exactly two events occurring) and
• P(Exactly three events occurring)

⇒ [P(P ∩ Q ∩ Rc) ∪ P(P ∩ Qc ∩ R) ∪ P(Pc ∩ Q ∩ R)] ∪ [P(P ∩ Q ∩ R)]

⇒ P(P ∩ Q ∩ Rc) + P(P ∩ Qc ∩ R) + P(Pc ∩ Q ∩ R) + P(P ∩ Q ∩ R)
[Since all these intersection events are mutually exclusive]

## Probability of Occurrence of atmost one or more of the two or more events

### Atmost one of the two events

For any two events occurrence of atmost one of the events implies the non occurrence of the event representing the intersection of those events.
• At most one of the events "A" and "B" ⇒ The complimentary of intersection of the events i.e. (A ∩ B)c

P(A ∩ B)c = P(Ac ∪ Bc)

### Atmost two of the three 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.
• At most two of the events "P", "Q" and "R" ⇒ The complimentary of intersection of the events i.e. (P ∩ Q ∩ R)c

P(P ∩ Q ∩ R)c = P(Pc ∪ Qc ∪ Rc)

### Atmost one of the three 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.
• At most one of the events "P", "Q" and "R"

⇒ P(P ∩ Qc ∩ Rc) + P(Pc ∩ Q ∩ Rc) + P(Pc ∩ Qc ∩ R) + P(Pc ∩ Qc ∩ Rc)

## Probability of Occurrence of only one or more of the two or more events

The use of the word only indicates the condition where only the mentioned event should occur and all other events should not occur.

### Only one of the two events

For two events "A" and "B" occurrence of only one of the events

⇒ P(A ∩ Bc) + P(Ac ∩ B)

### Only one of the three events

For three events "P", "Q", and "R" occurrence of only one of the events

⇒ P(P ∩ Qc ∩ Rc) + P(Pc ∩ Q ∩ Rc) + P(Pc ∩ Qc ∩ R)

### Only two of the three events

For three events "P", "Q", and "R" occurrence of only two of the events

⇒ P(P ∩ Q ∩ Rc) + P(P ∩ Qc ∩ R) + P(Pc ∩ Q ∩ R)

## Probability of Occurrence of all of the two or more events

The occurrence of all of the two or more events implies the occurrence of the intersection of the events

### All of the two events

For two events "A" and "B" occurrence of all of the events

⇒ P(A ∩ B)

### All of the three events

For three events "P", "Q", and "R" occurrence of all of the events

⇒ P(P ∩ Q ∩ R)

## Probability of Occurrence of none of the two or more events

The occurrence of none of the two or more events implies the occurrence of the intersection of the complimentaries of the individual events

### None of the two events

For two events "A" and "B" occurrence of none of the events

⇒ P(Ac ∩ Bc)

P(Ac ∩ Bc) = P(A ∪ B)c

### None of the three events

For three events "P", "Q", and "R" occurrence of none of the events

⇒ P(Pc ∩ Qc ∩ Rc)

P(Pc ∩ Qc ∩ Rc) = P(P ∪ Q ∪ R)c

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