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- Consider the probability space (Ω,A , P) where Ω = R and A is the Borel σ-algebra on R. Suppose that for any n = 1, 2, . . . , we have P((−∞, 2−n]) = 1/2 and P((−∞, −2−n]) = 1/3 − 1/(4^n) . Given the above, compute the following probability values, if possible, showing complete justifi- cation for every step. If it is impossible to compute a value precisely, provide the tightest possible bounds on it. P((−∞, 0]) P({0}) P((−∞, −1]) P((1/4, π/7]) limn→∞ P((0, n])Let ?, ?, ? ∈ Z. Suppose ?|?? and gcd(?, ?) = 1. Prove that ?|?. (Hint: use the following theorem proved in class: gcd(?, ?) = ?? + ?? for some ?, ?Find the splitting field x4 - x2 - 2 over Z3.