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Addition Theorem of Probability ➕

The Addition Theorem helps us calculate the probability of occurrence of at least one of two events. It answers: "What is the probability of obtaining A OR B?"

Symbolically: P(A ∪ B) or P(A or B).

Case 1: For Mutually Exclusive Events ⚡

If two events A and B cannot occur together (i.e., A ∩ B = 0), then:

P(A ∪ B) = P(A) + P(B)

Example: Mutually Exclusive ⚡

Tossing a die.

  • A: Getting 2. P(A) = 1/6.
  • B: Getting 5. P(B) = 1/6.
  • P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

Case 2: For Non-Mutually Exclusive Events 🤝

If A and B can occur together (they intersect), we must subtract the common part to avoid double counting.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Example: Non-Mutually Exclusive 🤝

Drawing a card.

  • A: King P(A) = 4/52.
  • B: Spade P(B) = 13/52.
  • Common (A ∩ B): King of Spades P(A ∩ B) = 1/52.
  • P(King or Spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.

Comparison 📊

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Theorem for 3 Events

For A, B, C (non-exclusive): P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) + P(A∩B∩C)

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