Problem 1 (25 pts) This problem consists of several separate…

Problem 1 (25 pts) This problem consists of several separate short questions relating to the structure of probability spaces: (a) Write the definition of a sigma field of a sample space \({\cal S}\). (b) Write down the axioms of probability. (c) If \(A\) and \(B\) are elements of a sigma field \({\cal F}\), show why \(A\cap B\) is also an element of \({\cal F}\). (d) From the axioms of probability, show that \(P(\emptyset)=0\). (e) From the axioms of probability, show that \(P(\overline{A})=1-P(A)\). Problem 2 (25 pts) Consider a probability space \(({\cal S},{\cal F},P)\).  Assume that \(A\in{\cal F}\) and \(B\in{\cal F}\).  Express each of the probabilities in (a) through (e) below in terms of \(P(A)\), \(P(B)\), and \(P(A\cap B)\).  In all cases, simplify as much as possible. (a) \(P(\overline{A}\cup \overline{B})\). (b) \(P(\overline{A}\cap \overline{B})\). (c) \(P(A\cup (\overline{A}\cap B))\). (d) \(P(\overline{A}| B)\). (e) \(P(A|\overline{B})\). (f) You are now told that (for part (f) only), \(P(A)=0.4\), \(P(A\cap B)=0.1\), and \(P(\overline{A\cup B})=0.2\). What is the numerical value of \(P(B)\)? Problem 3 (25 pts) Consider a random experiment with sample space S=0, 1, 2, 3…{“version”:”1.1″,”math”:”S=0, 1, 2, 3…”} and geometric probability mass function (pmf) $$p(k)=(1-a)a^k,\quad k=0,1,2,3,\ldots,$$ where \(0