Probability Addition Theorem Probability of At most, At least, Neither, All One or More Events
Addition Theorem of Probability
at least ⇔ Union
Two Non-Disjoint (Non-Mutually Exclusive) Events
For two events A and B are not disjoint (or not mutually exclusive),Probability that at least one of the events would occur
⇒ 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),Probability that at least one of the events would occur
⇒ 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 and Exhaustive Events
Experiment
Drawing a ball from a bag containing balls marked from 1 to 100
Where
S : Set of all elementary events in relation to the experiment
S = {1, 2, 3, 4, ... , 99, 100}
The union of sets of exhaustive events is the sample space i.e. the set of all elementary events in relation to the experiment.
Where
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
E ∪ O | = | {1, 2, 3, 4, ... , 99, 100} |
= | S |
Any events combined together with the exhaustive events (E and O here) would also form exhaustive events,
Since E and O together are exhaustive, any three or more events which include E and O 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
Sample Space - Probability of occurrence
Set of all elementary events (sample points) in relation to an experiment is the sample space. One or more of the elementary events (or sample points) occur on every iteration of the experiment.P(S) | = |
| ||
= | 1 |
⇒ Probability of occurrence of the sample space is a certainty.
exhaustive events - Probability
Since the union of exhaustive events is equal to the sample space, the probability of occurrence of the union of (at least one of the) exhaustive events is the same as the probability of the sample space i.e. 1.Two Exhaustive Events
For two events A and B which are exhaustive, the probability that at least 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 at least 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 and 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 happens when there are no sample points or elementary events common to the eventsIntersection of Mutually Exclusive Events
Since there are no common elements in the sets representing the events, the set representing the intersection of the events would be a null set.Mutually Exclusive Events - Probability of occurrence
The probability of occurrence of the intersection of mutually exclusive events is Nil.Examples
Two Events
WhereA = {1, 3, 5, .... (2n − 1) }
B = {2, 4, 6, .... (2n) }
A ∩ B = {} or Φ
⇒ n(A ∩ B) = 0
Events A and B are mutually exclusive.
P(A ∩ B) = n(A ∩ B) n(S) = 0 n(S) = 0 Three Events
WhereP = {a, b, c, .... m}
Q = {n, o, p, .... v}
R = {w, x, y, z}
P ∩ Q ∩ R = {} or Φ
⇒ n(P ∩ Q ∩ R) = 0
Events P, Q and R are mutually exclusive.
P(P ∩ Q ∩ R) = n(P ∩ Q ∩ R) n(S) = 0 n(S) = 0
Mutually Exclusive ⇒ Pair wise Exclusive
Where three or more events are in consideration, the events would be mutually exclusive if and only if they are pairwise mutually exclusive.Events A, B and C are mutually exclusive
only if
Events A and B are mutually exclusive
Events A and C are mutually exclusive
Events B and C are mutually exclusive
Probability of union of Disjoint (Mutually Exclusive) Events
The probability of occurrence of the union of (at least one of) two or more disjoint (mutually exclusive) events is given by the sum of the probabilities of the individual events.Two Events
For two events A and B which are disjoint (mutually exclusive),
P(A ∪ B)
= P(A) + P(B) − P(A ∩ B)
= P(A) + P(B) − 0
= P(A) + P(B)
Three Events
For three events E, F and G which are disjoint (mutually exclusive),
P(E ∪ F ∪ G)
= P(E) + P(F) + P(G) − P(E ∩ F) − P(E ∩ G) − P(F ∩ G) + P(E ∩ F ∩ G)
= P(E) + P(F) + P(G) − 0 − 0 − 0 + 0
= P(E) + P(F) + P(G)
Addition Theorem of Probability - Mutually Exclusive and Exhaustive Events
Two Events
For two events A and B which are mutually exclusive and exhaustive,P(A ∪ B) = P(A) + P(B)
Since they are mutually exclusive
P(A ∪ B) = 1
Since they are exhaustive
⇒ P(A ∪ B) = P(A) + P(B) = 1
Three Events
For three events A, B and C which are mutually exclusive and exhaustive,P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
Since they are mutually exclusive
P(A ∪ B ∪ C) = 1
Since they are exhaustive
⇒ P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1
Probability - Additional relations from set theory
Two Events
For any two events A and BP(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 CP(A ∪ B ∪ C)c = P(Ac ∩ Bc ∩ Cc)
P(A ∩ B ∩ C)c = P(Ac ∪ Bc ∪ Cc)
Occurrence of At least, At most, Only, None of the two or more events
At least
For two or more events occurrence of at least one of the events implies the occurrence of the event representing the union of those eventsat least ⇔ Union
At least one of the events A and B
⇒ The event A ∪ B
Additionally
A ∪ B | = | 1 − (A ∪ B)c |
= | 1 − (Ac ∩ Bc) |
At least one of the events P, Q and R
⇒ The event P ∪ Q ∪ R
Additionally
P ∪ Q ∪ R | = | 1 − (P ∪ Q ∪ R)c |
= | 1 − (Pc ∩ Qc ∩ Rc) |
At most
Two Events
For two or more events occurrence of at most one less than the number of events implies the non occurrence (complimentary) of the event representing the intersection of those events.At most one of the events A and B
⇒ The event (A ∩ B)c
Non occurrence (complimentary) of the event (A ∩ B)
Additionally
(A ∩ B)c = Ac ∪ Bc
Three Events
At most two of the events P, Q and R
⇒ The event (P ∩ Q ∩ R)c
Non occurrence (complimentary) of the event (P ∩ Q ∩ R)
Additionally
(P ∩ Q ∩ R)c = Pc ∪ Qc ∪ Rc
At most one of the events P, Q and R
⇏ The event (P ∩ Q ∩ R)c
⇒ 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.Two Events
Only one of the events A and B
⇒ P(A ∩ Bc) + P(Ac ∩ B)
Three Events
Only one of the events P, Q and R
⇒ P(P ∩ Qc ∩ Rc) + P(Pc ∩ Q ∩ Rc) + P(Pc ∩ Qc ∩ R)
Only two of the events P, Q and R
⇒ P(P ∩ Q ∩ Rc) + P(P ∩ Qc ∩ R) + P(Pc ∩ Q ∩ R)