Alright, let's break down the problem and determine the outcomes for each event step-by-step:
1. First, let's identify the events A and B:
- Event [tex]\( A \)[/tex]: The marble selected has an odd number.
- Event [tex]\( B \)[/tex]: The marble selected has a number greater than 3.
2. Step 1: List all possible marbles outcomes:
The bag contains marbles numbered from 1 to 8. Therefore, the sample space [tex]\( S \)[/tex] is:
[tex]\( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \)[/tex]
3. Step 2: Determine the outcomes for Event [tex]\( A \)[/tex]:
Event [tex]\( A \)[/tex] implies that the marble has an odd number.
[tex]\[
A = \{1, 3, 5, 7\}
\][/tex]
4. Step 3: Determine the outcomes for Event [tex]\( B \)[/tex]:
Event [tex]\( B \)[/tex] implies that the marble has a number greater than 3.
[tex]\[
B = \{4, 5, 6, 7, 8\}
\][/tex]
5. Step 4: Determine the outcomes for Event " [tex]\( A \)[/tex] and [tex]\( B \)[/tex]":
Event " [tex]\( A \)[/tex] and [tex]\( B \)[/tex]" implies that the marble has both an odd number and a number greater than 3.
We find the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A \cap B = \{5, 7\}
\][/tex]
Thus, the outcomes for Event " [tex]\( A \)[/tex] and [tex]\( B \)[/tex]" are:
[tex]\[
\{5, 7\}
\][/tex]
So, the answer to (a) is:
[tex]\[
\{5, 7\}
\][/tex]
6. Step 5: Determine the outcomes for Event " [tex]\( A \)[/tex] or [tex]\( B \)[/tex]":
Event " [tex]\( A \)[/tex] or [tex]\( B \)[/tex]" implies that the marble can either have an odd number or a number greater than 3 or both.
We find the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A \cup B = \{1, 3, 4, 5, 6, 7, 8\}
\][/tex]
Thus, the outcomes for Event " [tex]\( A \)[/tex] or [tex]\( B \)[/tex]" are:
[tex]\[
\{1, 3, 4, 5, 6, 7, 8\}
\][/tex]
So, the answer to (b) is:
[tex]\[
\{1, 3, 4, 5, 6, 7, 8\}
\][/tex]
7. Step 6: Determine the outcomes for the complement of the Event [tex]\( B \)[/tex]:
The complement of Event [tex]\( B \)[/tex] implies that the marble has a number not greater than 3 (i.e., less than or equal to 3).
[tex]\[
B^c = \{1, 2, 3\}
\][/tex]
Thus, the outcomes for the complement of Event [tex]\( B \)[/tex] are:
[tex]\[
\{1, 2, 3\}
\][/tex]
So, the answer to (c) is:
[tex]\[
\{1, 2, 3\}
\][/tex]
In summary:
(a) The outcomes for Event " [tex]\( A \)[/tex] and [tex]\( B \)[/tex]" are: \{5, 7\}
(b) The outcomes for Event " [tex]\( A \)[/tex] or [tex]\( B \)[/tex]" are: \{1, 3, 4, 5, 6, 7, 8\}
(c) The outcomes for the complement of Event [tex]\( B \)[/tex] are: \{1, 2, 3\}
