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Four speed skaters, Marco, Naim, Oliver, and Pedro, compete in a relay race where they all agree that Naim will be the last skater. Then, they try to decide whether or not Oliver should be the first skater in the race.

Which subset, A, of the sample space shows the complement of the event in which Oliver is the first skater in the race?

A = {MOPN, PMON}
A = {OPMN, OMPN}
A = {MOPN, MPON, OPMN, POMN}
A = {MOPN, MPON, PMON, POMN}

1 answer

To solve this problem, we start by identifying the constraints given:

  1. Naim must be the last skater.
  2. We are interested in the situation where Oliver is not the first skater.

Since Naim will always be last, we only need to arrange the three skaters (Marco, Oliver, and Pedro) for the first three positions.

The sample space without any constraints (for the first three skaters, followed by Naim) would be:

  1. MOPN
  2. MONP
  3. OMPN
  4. OMPN
  5. PMON
  6. POMN

Next, we need to identify the outcomes where Oliver is the first skater. The relevant arrangements with Oliver first are:

  1. OMPN
  2. OPMN

Now, the complement of the event where Oliver is the first skater consists of all outcomes that do not have Oliver in the first position.

Looking at the possible sequences, if we eliminate the outcomes where Oliver is first, we are left with:

  1. MOPN
  2. MONP
  3. PMON
  4. POMN

Now we can examine the options provided in the question:

  1. A = {MOPN, PMON} - This does not include all outcomes where Oliver is not in the first position.
  2. A = {OPMN, OMPN} - This does not represent the complement since it contains cases with Oliver first.
  3. A = {MOPN, MPON, OPMN, POMN} - This set contains more than the correct outcomes (MPON and OPMN includes Oliver first).
  4. A = {MOPN, MPON, PMON, POMN} - This set also includes outcomes where Oliver is not first with the incorrect inclusion of MPON (where Oliver could be present).

None of the provided options correspond exactly to the complement defined by the inclusion/exclusion of Oliver first.

To provide the most accurate response based upon eliminating all outcomes with Oliver first, we conclude:

The correct subset A showing the complement event of Oliver not being the first is:

A = {MOPN, MPON, PMON, POMN}

This option includes all combinations of Marco, Pedro, and Naim, without Oliver leading the race.