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The addition theorem in the Probability concept is the process of determination of the probability that either event ‘A’ or event ‘B’ occurs or both occur. The notation between two events ‘A’ and ‘B’ the addition is denoted as '∪' and pronounced as Union.
The result of this addition theorem generally written using Set notation,
P (A ∪ B) = P(A) + P(B) – P(A ∩ B),
Where,
P (A) = probability of occurrence of event ‘A’
P (B) = probability of occurrence of event ‘B’
P (A ∪ B) = probability of occurrence of event ‘A’ or event ‘B’.
P (A ∩ B) = probability of occurrence of event ‘A’ or event ‘B’.
Addition theorem probability can be defined and proved as follows:
Let ‘A’ and ‘B’ are Subsets of a finite non empty set ‘S’ then according to the addition rule
P (A ∪ B) = P (A) + P (B) – P (A). P (B),
On dividing both sides by P(S), we get
P (A ∪ B) / P(S) = P (A) / P(S) + P (B) / P(S) – P (A ∩ B) / P(S) (1).
If the events ‘A’ and ‘B’ correspond to the two events ‘A’ and ‘B’ of a random experiment and if the set ‘S’ corresponds to the Sample Space ‘S’ of the experiment then the equation (1) becomes
P (A ∪ B) = P (A) + P (B) – P (A). P (B),
This equation is known as the addition theorem in probability.
Here the event A ∪ B refers to the meaning that either event ‘A’ or event ‘B’ occurs or both may occur simultaneously.
If two events A and B are Mutually Exclusive Events then A ∩ B = ф,
Therefore
P (A ∪ B) = P (A) + P(B) [since P(A ∩ B) = 0],
In language of set theory A ∩ B̅ is same as A / B.

Define Addition Theorem

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Probability is the branch of mathematics, which is concerned with analysis of random phenomena. In simple words, probability is certainty of occurring of an event or how likely an event can occur. We can predicate the total certainty of an event in real world. In Probability we use few things which are defined as central objects of probability like Random Variable, stochastic processes and events.
For understanding probability one can take example of tossing a coin. When we tossed any coin there is only two possible outcomes i.e. either we get head or tail. In such case probability is 50 – 50 means ½ - ½. There are different type of events in probability like Independent, Mutually Exclusive, Conditional, and compound events.
Now, turn gear towards other important topic of probability that is Addition Theorem of probability. Before we define addition theorem of probability it is necessary to understand compound event. When we combine two or more simple events then the resultant event is called as compound event. In this case, we use this addition theorem.

The addition theorem of probability states that, when we have to find that event 'A' has occurred or 'B' then we find total possible number of ways in which 'A' can occur and total number of ways in which 'B' can occur.
In such a way, that each outcome is counted once but not more than once.

The notation for addition rule is: P (A∪B) = P (A) + P (B) – P (A∩B)
Here, U = or and ∩ = and. It notation says that A and B both will occur at the same time as an outcome in a trial procedure.
This is a brief introduction about probability and addition theorem of probability.