Answer :
Given four speed skaters, Marco, Oliver, Pedro, and Naim, it's agreed that Naim will be the last skater in the race. We need to find the complement of the event where Oliver is the first skater in the race.
First, we list all possible orders where Naim is the last skater:
- MOPN
- MONP
- MPO N
- MPNO
- OMPN
- OMNP
- OPMN
- OPNM
- ONMP
- ONPM
- PMON
- PMNO
- PNOM
- PONM
- POMN
- PONM
Next, we identify the permutations where Oliver is the first skater:
- OMPN
- OMNP
- OPMN
- OPNM
- ONMP
- ONPM
The complement of this event is the set of all permutations without Oliver being the first skater. Therefore, we remove all permutations where Oliver is the first skater from the list. This leaves us with:
- MOPN
- MONP
- MPON
- MPNO
- PMON
- PMNO
- PNOM
- PONM
- POMN
- PONM
So, the subset [tex]\( A \)[/tex] of the sample space that shows the complement of the event in which Oliver is the first skater in the race is:
[tex]\[ A = \{ MOPN, MONP, MPON, MPNO, PMON, PMNO, PNOM, PONM, POMN \} \][/tex]
Comparing this to the provided options:
[tex]\[ A = \{MOPN, MPON, PMON, POMN\} \][/tex] matches correctly.
Therefore, the correct answer is:
[tex]\[ A=\{MOPN, MPON, PMON, POMN\} \][/tex]
First, we list all possible orders where Naim is the last skater:
- MOPN
- MONP
- MPO N
- MPNO
- OMPN
- OMNP
- OPMN
- OPNM
- ONMP
- ONPM
- PMON
- PMNO
- PNOM
- PONM
- POMN
- PONM
Next, we identify the permutations where Oliver is the first skater:
- OMPN
- OMNP
- OPMN
- OPNM
- ONMP
- ONPM
The complement of this event is the set of all permutations without Oliver being the first skater. Therefore, we remove all permutations where Oliver is the first skater from the list. This leaves us with:
- MOPN
- MONP
- MPON
- MPNO
- PMON
- PMNO
- PNOM
- PONM
- POMN
- PONM
So, the subset [tex]\( A \)[/tex] of the sample space that shows the complement of the event in which Oliver is the first skater in the race is:
[tex]\[ A = \{ MOPN, MONP, MPON, MPNO, PMON, PMNO, PNOM, PONM, POMN \} \][/tex]
Comparing this to the provided options:
[tex]\[ A = \{MOPN, MPON, PMON, POMN\} \][/tex] matches correctly.
Therefore, the correct answer is:
[tex]\[ A=\{MOPN, MPON, PMON, POMN\} \][/tex]