Let's consider the given events:
Event [tex]\(A\)[/tex]: The number rolled is less than 5.
Event [tex]\(B\)[/tex]: The number rolled is odd.
Based on these events, we can determine the outcomes for each question:
(a) Event "[tex]$A$[/tex] or [tex]$B$[/tex]":
This event includes all outcomes that are either in event [tex]\(A\)[/tex] or in event [tex]\(B\)[/tex]. We need to find the union of events [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Event [tex]\(A\)[/tex] (numbers less than 5) consists of: [tex]\(\{1, 2, 3, 4\}\)[/tex].
Event [tex]\(B\)[/tex] (odd numbers) consists of: [tex]\(\{1, 3, 5\}\)[/tex].
The union of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is:
[tex]\[
A \cup B = \{1, 2, 3, 4, 5\}
\][/tex]
(b) Event "[tex]$A$[/tex] and [tex]$B$[/tex]":
This event includes all outcomes that are in both event [tex]\(A\)[/tex] and event [tex]\(B\)[/tex]. We need to find the intersection of events [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
From above, we have:
Event [tex]\(A\)[/tex] (numbers less than 5): [tex]\(\{1, 2, 3, 4\}\)[/tex].
Event [tex]\(B\)[/tex] (odd numbers): [tex]\(\{1, 3, 5\}\)[/tex].
The intersection of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is:
[tex]\[
A \cap B = \{1, 3\}
\][/tex]
(c) The complement of event [tex]\(B\)[/tex]:
This event includes all outcomes that are not in event [tex]\(B\)[/tex]. We need to find the complement of event [tex]\(B\)[/tex].
From above, event [tex]\(B\)[/tex] is [tex]\(\{1, 3, 5\}\)[/tex].
The set of all possible outcomes when rolling a number cube labeled 1 to 6 is: [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex].
The complement of event [tex]\(B\)[/tex] is:
[tex]\[
B^C = \{ 1, 2, 3, 4, 5, 6 \} \setminus \{1, 3, 5\} = \{2, 4, 6\}
\][/tex]
So, summarizing the answers:
(a) Event "[tex]$A$[/tex] or [tex]$B$[/tex]": [tex]\(\{1, 2, 3, 4, 5\}\)[/tex]
(b) Event "[tex]$A$[/tex] and [tex]$B$[/tex]": [tex]\(\{1, 3\}\)[/tex]
(c) The complement of the event [tex]\(B\)[/tex]: [tex]\(\{2, 4, 6\}\)[/tex]
