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

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Addition Theorem of Probability

 
 

• 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 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.
  • A = {1, 3, 5, .... (2n − 1) }
  • B = {2, 4, 6, .... (2n) }
Events "A" and "B" are said to be mutually exclusive.

» Three Events

Three events are mutually exclusive only if they do not have an element in common.
  • P = {a, b, c, .... m}
  • Q = {n, o, p, .... v}
  • R = {w, x, y, z}
Events "P", "Q" and "R" are said to be mutually exclusive.

• 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.

Thus P(A ∩ B) =
n(A ∩ B)
n(S)
and P(P ∩ Q ∩ R) =
n(P ∩ Q ∩ R)
n(S)
=
0
n(S)
=
0
n(S)
= 0 = 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

  • S = {1, 2, 3, 4, ... , 99, 100}
  • E = {2, 4, 6, 8, ... , 98, 100}
  • O = {1, 3, 5, 7, ... , 97, 99}
  • F = {5, 10, ... , 95, 100}
  • T = {3, 6, 9, 12, ... , 96, 99}

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,

  • "E", "O" and "F" are exhaustive events ⇒ E ∪ O ∪ F = S.
  • "E", "O" and "T" are exhaustive events ⇒ E ∪ O ∪ T = S
  • "E", "O", "F" and "T" are exhaustive events ⇒ E ∪ O ∪ F ∪ T = S

Probability of occurrence of the sample space is a certainty i.e. its probability is 1.
P(S) =
n(S)
n(S)
= 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.

P(A ∪ B) = P(A) + P(B)[Mutually Exclusive]
P(A ∪ B) = 1[Exhaustive]
P(A ∪ B) = P(A) + P(B) = 1[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]
P(A ∪ B ∪ C) = 1[Exhaustive]
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1[Mutually Exclusive and Exhaustive]

Probability » Additional relations 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)

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

• 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.
  • At most one of the events "A" and "B" ⇒ The complimentary of intersection of the events i.e. (A ∩ B)c
    (A ∩ B)c = Ac ∪ Bc

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 ∩ Q ∩ R)c = Pc ∪ Qc ∪ Rc

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)

• 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
⇒ P(A ∩ Bc) + P(Ac ∩ B)

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)

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)

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