Theorem2
Let us take A and B be the events that are two nonmutually exclusive then,
P(A ∪ B)=P(A) + P(B)  P(A ∩ B).
Proof
Let us take p = total number of exhaustive cases
p_{1}= number of cases favorable to A.
p_{2}= number of cases favorable to B.
p_{3}= number of cases favorable to both A and B.
Since A and B are not mutually exclusive as a result, A and B can occur at the same time. Therefore, the number of situations favourable to A or B will be p_{1 }+ p_{2 } p_{3}.
Therefore, P(A ∪ B) = (p_{1} + p_{2 } p_{3}) ⁄ p = p_{1 }⁄ p + p_{2} ⁄_{ }p  p_{3} ⁄_{ }p
Since we have,
P(A) = p_{1} ⁄ p
P(B) = p_{2 }⁄_{ }p
P(A ∩ B) = p_{3 }⁄ p
Hence, P(A ∪ B)=P(A) + P(B)  P(A ∩ B).
Examples

If P(A) = 0.27 , P(B) = 0.39 , P (A ∩ B) = 0.06 then find P (A U B) .
Solution:
P(A) = 0.27 , P(B) = 0.39 , P (A ∩ B) = 0.06
P (A U B) = P (A) + P (B ) − P (A ∩ B)
P (A U B) = 0.27 + 0.39 − 0.06 = 0. 6

Find out the probability of drawing either a king or a queen in a single draw from a well shuffled pack of 52 cards.
Solution: We have,
Total number of cards = 52
Number of queens in a deck = 4
Probability of drawing a queen = 4/52
Number of kings in the deck = 4
Probability of drawing a king = 4/52
Since, Both the events of drawing a king and a queen are mutually exclusive
Therefore, P (A ∪ B) = P (A) + P (B)
= 4/52 + 4/52 = 2/13
Therefore, the probability of drawing either a king or a queen is 2/13.
FAQs

What do you mean by a mutually exclusive event?
Events that do not occur simultaneously are known as mutually exclusive events. For example, When a coin is tossed, the outcome will be either head or tail, but we cannot obtain both. Because such events do not occur at the same time, they are referred to as discontinuous events.

How do you know if events are mutually exclusive?
When two events (say A and B) have no elements in common, their intersection becomes an empty set. Therefore, these types of events are called mutually exclusive. Thus, P(A∩B)=0 . This means that event A and event B's probability is zero.
Key Takeaways
In this article we have extensively discussed the Addition Theorem Of Probability with the help of examples. Check out the Conditional Probability for further topics.
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