Suppose that b1 b2nbspomega isin omega is a sequence of

Question

The problem is in attachment. Please elaborate each steps as detailed as possible. Thanks.

Consider a probability space (Ω, F, Ρ) such that Ω is countable and F = 2^Ω.

(i) Suppose that B₁, B₂,......... ω ∈ Ω is a sequence of independent events such that for all i, P[B_i] = 1/2. Show that for any n ≥ 1 and for any ω ∈ Ω, we have

P[{ω}] ≤ 1/2ⁿ

(ii) Using the result of part (i), show that it is impossible for such a sequence B₁, B₂, ......... ∈ F to exist!