Let Ω={1,2,3,4,5,6}\Omega = \{1,2,3,4,5,6\} and ℱ={∅,Ω,{1,3,…
Let Ω={1,2,3,4,5,6}\Omega = \{1,2,3,4,5,6\} and ℱ={∅,Ω,{1,3,5},{2,4,6}}\mathcal{F} = \{\emptyset,\ \Omega,\ \{1,3,5\},\ \{2,4,6\}\} be as in Question 1, and define P(A)=|A|/6P(A) = |A|/6 for A∈ℱA \in \mathcal{F}, where |A||A| denotes the number of elements of AA. Verify that PP satisfies the three Kolmogorov axioms. (i) (2 points) P(A)≥0P(A) \geq 0 for every A∈ℱA \in \mathcal{F}. (ii) (2 points) P(Ω)=1P(\Omega) = 1. (iii) (2 points) P(⋃i=1∞Ai)=∑i=1∞P(Ai)P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i) whenever A1,A2,…∈ℱA_1, A_2, \ldots \in \mathcal{F} are pairwise disjoint.
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