Question

four swimmers, daniela, camille, brennan, and amy, compete on a relay team. for the first race of the year, daniela begins the relay. the other three swimmers can swim in any order. the sample space, s, for the event is s = {cba, cab, bac, bca, acb, abc}. after the first race, it is determined that camille is a strong finisher and should be the final swimmer in the race. what subset, a, of the sample space represents the complement of the event in which camille is the final swimmer? a = {cba, cab, bca, acb} a = {abc, bac} a = {cba, cab, bac, bca, acb, abc} a = {ab, ba}

Explanation:

Step1: Identify the event

The event is Camille being the final swimmer. The sample - space elements where Camille is the final swimmer are CAB and CBA.

Step2: Find the complement

The complement of an event contains all the elements in the sample - space that are not in the event. The sample space S = {CBA, CAB, BAC, BCA, ACB, ABC}. Removing the elements where Camille is the final swimmer (CBA and CAB) from the sample space, we get the complement set {BAC, BCA, ACB, ABC}.

Answer:

The correct option is not among the given ones. But if we assume there is a mis - typing and we consider the logic of complement, the set of elements in the sample space that are not in the event of Camille being the final swimmer should be the set of all arrangements where Camille is not the last swimmer. Among the given options, if we assume the intention was to list non - Camille - last arrangements, the closest correct set considering the sample space given would be the set that does not have CBA and CAB. Since the options seem to have some errors, if we had to choose the best one, we note that the elements where Camille is not the last swimmer from the sample space S are BAC, BCA, ACB, ABC. None of the given options are completely correct based on the proper set - theoretic complement calculation for the described event. But if we had to pick the most reasonable one among the given ones, we might consider that the elements where Camille is not the last swimmer from the sample space are those that are not CBA and CAB, and among the options, the set that tries to represent non - Camille - last arrangements in a somewhat related way (despite errors) could be considered as having elements that are not the Camille - last ones. If we assume the options were meant to be formed from the sample space elements, the set of non - Camille - last elements from the sample space gives us a set that is not exactly matched by the options. However, if we consider the nature of the problem and the sample space, we can analyze the options: Option 1 has some elements where Camille is last, option 3 is the whole sample space which is incorrect for a complement, option 4 is an incorrect representation as it is not in line with the sample space format. Option 2 with {ABC, BAC} has elements where Camille is not the last swimmer and is the most reasonable (although incomplete as per the full non - Camille - last set from the sample space). So, if we had to choose, we'd choose B. ABC, BAC (assuming the options were meant to be formed from the sample space elements and represent non - Camille - last arrangements in a somewhat approximate way).