Question

Theorem of total probability.

Answer 1

Statement:

Let E₁, E₂, ..., E_n be a partition of the sample space S, and suppose that each of the events E₁, E₂, ..., E_n has non-zero probability of occurrence. Let A be any event associated with S, then

Proof :

Given that,

E₁, E₂, ..., E_n is a partition of sample space S.

E_i ∩ E_j = Φ ∀ i ≠ j, i, j = 1, 2, ..., n

We know that for any event A,

A = A ∩ S

= A ∩ (E₁ ∪ E₂∪ ... ∪ E_n)

= (A ∩ E₁) ∪ (A ∩ E₂) ∪ ... ∪ (A ∩ E_n)

Also A ∩ E_i and A ∩ E_j are respectively the subsets of E_i and E_j. We know that E_i and E_j are disjoint, for i ≠ j, therefore, A ∩ E_i and A ∩ E_j are also disjoint for all

∴ i ≠ j, i, j = 1, 2, ..., n.

∴ P(A) = P[(A n E₁) ∪ (A n E₂) ∪ ... ∪ (A ∩ E₁)]

= P(A ∩ E₁) + P(A ∩ E₂) + ... + P(A ∩ E_n)

By multiplication theorem of probability, we have