Let's break down the problem and look at each event step by step.
Event [tex]$A$[/tex]: The marble selected has a number less than 4.
The numbers on the marbles in this event are: {1, 2, 3}
Event [tex]$B$[/tex]: The marble selected has an even number.
The numbers on the marbles in this event are: {2, 4, 6, 8}
Now, let's determine the outcomes for each specified event:
### Part (a) Event \"[tex]$A$[/tex] and [tex]$B$[/tex]\":
We need to find the marbles that satisfy both event [tex]$A$[/tex] and event [tex]$B$[/tex]. In other words, we are looking for numbers that are both less than 4 and even. From the events, we get:
- Event [tex]$A$[/tex] numbers: {1, 2, 3}
- Event [tex]$B$[/tex] numbers: {2, 4, 6, 8}
The intersection of these sets (common elements) is: {2}.
Outcome for event " [tex]$A$[/tex] and [tex]$B$[/tex] " is {2}.
### Part (b) Event \"[tex]$A$[/tex] or [tex]$B$[/tex]\":
We need to find the marbles that satisfy either event [tex]$A$[/tex] or event [tex]$B$[/tex]. This includes all numbers that are either less than 4 or even. To do this, we take the union of both events:
- Event [tex]$A$[/tex] numbers: {1, 2, 3}
- Event [tex]$B$[/tex] numbers: {2, 4, 6, 8}
Combining these sets (without repeating elements) gives us: {1, 2, 3, 4, 6, 8}.
Outcome for event " [tex]$A$[/tex] or [tex]$B$[/tex] " is {1, 2, 3, 4, 6, 8}.
### Part (c) The complement of the event [tex]$B$[/tex]:
The complement of event [tex]$B$[/tex] consists of all marbles that are not in event [tex]$B$[/tex]. This means we want numbers that are not even (i.e., odd numbers).
- Event [tex]$B$[/tex] numbers: {2, 4, 6, 8}
The numbers in the total set {1, 2, 3, 4, 5, 6, 7, 8} that are not in event [tex]$B$[/tex] are: {1, 3, 5, 7}.
Outcome for the complement of event [tex]$B$[/tex] is {1, 3, 5, 7}.
To summarize:
(a) Event " [tex]$A$[/tex] and [tex]$B$[/tex] " is {2}.
(b) Event " [tex]$A$[/tex] or [tex]$B$[/tex] " is {1, 2, 3, 4, 6, 8}.
(c) The complement of the event [tex]$B$[/tex] is {1, 3, 5, 7}.
