To solve this problem, let's first understand the events and then determine the desired outcomes step-by-step.
Given:
- Event [tex]\( A \)[/tex]: The number rolled is greater than 4.
- Possible outcomes for [tex]\( A \)[/tex]: \{5, 6\}
- Event [tex]\( B \)[/tex]: The number rolled is even.
- Possible outcomes for [tex]\( B \)[/tex]: \{2, 4, 6\}
Now, let's determine the outcomes for each specified event:
(a) Event " [tex]\( A \)[/tex] or [tex]\( B \)[/tex] ": This event includes all outcomes that are either in [tex]\( A \)[/tex] or in [tex]\( B \)[/tex]. Essentially, this is the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- [tex]\( A \cup B \)[/tex] = \{5, 6\} ∪ \{2, 4, 6\} = \{2, 4, 5, 6\}
- So, the outcome for this event is: [tex]\(\{2, 4, 5, 6\}\)[/tex]
(b) Event " [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ": This event includes only the outcomes that are in both [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Essentially, this is the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- [tex]\( A \cap B \)[/tex] = \{5, 6\} ∩ \{2, 4, 6\} = \{6\}
- So, the outcome for this event is: [tex]\(\{6\}\)[/tex]
(c) The complement of event [tex]\( B \)[/tex]: This event includes all outcomes that are not in [tex]\( B \)[/tex].
- The sample space (all possible outcomes on a number cube) is: \{1, 2, 3, 4, 5, 6\}
- Complement of [tex]\( B \)[/tex] (not in [tex]\( B \)[/tex]): \{1, 2, 3, 4, 5, 6\} - \{2, 4, 6\} = \{1, 3, 5\}
- So, the outcome for this event is: [tex]\(\{1, 3, 5\}\)[/tex]
Thus, the resulting outcomes for each event are:
(a) Event " [tex]\( A \)[/tex] or [tex]\( B \)[/tex]": [tex]\( \{2, 4, 5, 6\} \)[/tex]
(b) Event " [tex]\( A \)[/tex] and [tex]\( B \)[/tex]": [tex]\( \{6\} \)[/tex]
(c) The complement of the event [tex]\( B \)[/tex]": [tex]\( \{1, 3, 5\} \)[/tex]
