To solve this problem, let's break it down step-by-step.
1. Sample Space (S):
- The sample space consists of all possible pairs of slips that can be drawn simultaneously.
- Given slips are marked with the numbers 3, 4, 5, 6, and 7.
- The pairs are formed by picking two different numbers from these slips.
- The possible pairs are:
[tex]\[
(3, 4), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)
\][/tex]
Therefore, the sample space [tex]\(S\)[/tex] is:
[tex]\[
S = \{(3, 4), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)\}
\][/tex]
2. Value of [tex]\(n(S)\)[/tex]:
- [tex]\(n(S)\)[/tex] represents the number of outcomes in the sample space.
- Counting the number of pairs listed above, we get:
[tex]\[
n(S) = 10
\][/tex]
3. Equally Likely Outcomes:
- Since the slips are drawn simultaneously and each slip is equally likely to be picked, the outcomes in [tex]\(S\)[/tex] are equally likely.
4. Events in Set Notation:
a. Both slips are marked with odd numbers:
- The odd numbers among the slips are 3, 5, and 7.
- Possible pairs of odd numbers are:
[tex]\[
(3, 5), (3, 7), (5, 7)
\][/tex]
Therefore, the set notation for event [tex]\(A\)[/tex] is:
[tex]\[
A = \{(3, 5), (3, 7), (5, 7)\}
\][/tex]
b. One slip is marked with an odd number and the other is marked with an even number:
- The odd numbers are 3, 5, 7 and the even numbers are 4, 6.
- Possible pairs where one slip is odd and the other is even are:
[tex]\[
(3, 4), (3, 6), (4, 5), (4, 7), (5, 6), (6, 7)
\][/tex]
Therefore, the set notation for event [tex]\(B\)[/tex] is:
[tex]\[
B = \{(3, 4), (3, 6), (4, 5), (4, 7), (5, 6), (6, 7)\}
\][/tex]
c. Both slips are marked with the same number:
- Since all numbers on the slips are distinct, it is not possible to draw two slips with the same number.
Therefore, the set notation for event [tex]\(C\)[/tex] is:
[tex]\[
C = \{\}
\][/tex]
In summary:
- The correct sample space is:
[tex]\[
S = \{(3, 4), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)\}
\][/tex]
- The value of [tex]\(n(S)\)[/tex] is 10.
- The outcomes in [tex]\(S\)[/tex] are equally likely.
- The sets representing the indicated events are:
[tex]\[
A = \{(3, 5), (3, 7), (5, 7)\}
\][/tex]
[tex]\[
B = \{(3, 4), (3, 6), (4, 5), (4, 7), (5, 6), (6, 7)\}
\][/tex]
[tex]\[
C = \{\}
\][/tex]
