Question

Consider an experiment corresponding to the single throw of a die. Let E1 be the event that the throw is one of 11,2,3,4) and let E2 the event that the throw is one of 13,4,5,6). You are given that P(E) P(E2) 2/3. (a) Identify the smallest o-algebra sf containing the events E1 and E2. (b) Attach probabilities P(J) to all the elements J E (c) What can you say about the probability P(11]) of the event associated to a throw of 1? Be as precise as you can be.

Answer 1

$(a)~Let~A=smallest~\sigma-algebra\\ A=\{\phi,~\Omega,\{1,2,3,4\},\{3,4,5,6\},\{1,2\},\{5,6\},\{1,2,5,6\},\{3,4\}\}\\ Check~that~\{1,2,3,4\}^c=\{5,6\},~\{3,4,5,6\}^c=\{1,2\},\\ \{1,2\}\cup\{5,6\}=\{1,2,5,6\},~\{1,2,5,6\}^c=\{3,4\}\\ (b)~Attach:~P(\{1,2\})=P(\{3,4\})=P(\{5,6\})=\frac{1}{3}\\ then~P(\{1,2,3,4\})=P(\{1,2\})+P(\{3,4\})=2/3\\ ~since~\{1,2\}~and~\{3,4\}~are~disjoint~and~use~axiom~of~additivity\\ Similarly,~P(\{1,2,5,6\})=P(\{3,4,5,6\})=2/3\\ (c)~Since~\{1\}~is~not~a~element~of~A~so~we~can't~define~P(\{1\})~on~A.\\$