Addition Theorem

Theorem-2

Let us take A and B be the events that are two non-mutually exclusive then,

P(A ∪ B)=P(A) + P(B) - P(A ∩ B).

Proof

Let us take p = total number of exhaustive cases

              p₁= number of cases favorable to A.

              p₂= number of cases favorable to B.

              p₃= 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 p₁ + p₂ - p₃.

Therefore,  P(A ∪ B) =  (p₁ + p₂ - p₃)  ⁄  p = p₁ ⁄ p + p₂ ⁄ p - p₃ ⁄ p

Since we have, 

P(A) = p₁ ⁄ p 

P(B) = p₂ ⁄ p 

P(A ∩ B) =  p₃ ⁄ p

Hence, P(A ∪ B)=P(A) + P(B) - P(A ∩ B).

 

Examples

1. 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
    
2. 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

1. 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.
    
2. 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.

Recommended Readings:

• Initial Value Theorem
• 8 Queens Problem Using Backtracking

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