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

Hello Fresh Inc. is in the healthy foods home delivery busin…

Hello Fresh Inc. is in the healthy foods home delivery business. In each of their markets, they aim to provide every household with wholesome, homemade meals – no shopping and no hassle. In 2022, they want to increase overall sales by 20% and customer satisfaction by 10%.  As the HR Business Partner in the Sales Division, describe two (2) ways that you could contribute to achieving the sales and/or customer satisfaction goals for Hello Fresh.  Remember, you are in HR, and you are not selling the products/services or handling customer issues.