1.

Let a pair of fair coins be tossed. Here S = {HH, HT, TH, TT}. Consider the events A = {heads on the first coin} = {HH, HT}, B = {heads on the second coin} = {HH, TH}, C = {heads on exactly one coin} = {HT, TH}

Answer»

Then

P (A) = P (B) = P (C) = \(\frac{2}{4}\) = \(\frac{1}{2}\) and

P (A ∩ B) = P ({HH}) = \(\frac{1}{4}\), P (A ∩ C) = P ({HT}) = \(\frac{1}{4}\)

P (B ∩ C) = P ({TH}) = \(\frac{1}{4}\), (A ∩ B ∩ C) = ϕ

∴ P (A ∩ B ∩ C) = P (ϕ) = 0 ≠ P (A). P (B). P (C)

Thus condition (i) is satisfied, i.e., the events are pairwise independent. But condition (ii) is not satisfied and so the three events are not independent



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