The answer can best be described as indeterminate.
The difficulty in this problem is that it is self-referential and appears at first observation to be self-contradictory.
However, there are two hidden assumptions contained within the problem. These are worth examining and in doing so there does appear to be a solution
that is to be preferred over others.
The first assumption is that “random selection” implies that each of the four answers may be chosen with equal probability. This is a natural
interpretation of the phrase “choose a random answer”, but not a necessary interpretation. What has to be acknowledged is that whenever a random
selection is required, there must be some kind of random selection process. I might put A, B, C and D on a dart board and use my awesome throwing
skills to make a selection. The selection will be random, that is, subject to chance. But there is nothing to suggest all four outcomes have equal
probability.
I might, with equal validity use a regular six-sided dice and label the faces A, B, B, B, C, D. This would give me a random choice and in this case
the probability of selecting B is 50% which incidentally matches that question option. Thus B could be a correct answer.
Alternatively, I might label an eight-sided dice with A, C, C, C, C, C, C, D. In this case there is a 25% chance of obtaining A or D. Both A and D
state a figure of 25% which matches this probability. Therefore, with this random process, either A or D could be considered the correct solution.
And it would not matter which of the two I chose.
A different six-sided dice could be labelled A, A, B, B, D, D. With this random selection there is 0% probability of obtaining C; which matches
option C. Therefore, C could be a correct answer.
And as it has been pointed out ably by Metacelsus, a perfectly uniform random selection process necessarily leads to a paradox and therefore no
sensible answer at all. There are plenty of random selection processes that lead to such paradoxes: this is not the only one.
(And then there is the reverse paradox. If I select randomly using an eight-sided dice labelled A, B, B, B, B, E, E, D then I could make a case that
all the answers are correct since the probabilities of their selection match the numerical answers provided. This opens up the contradiction that
there is 100% chance of obtaining a correct answer in spite of the fact that 100% is not an option. There is also a second paradox under this scheme
in that contradictory answers should be considered equally true. I find this paradoxical situation even more bizarre than the uniform selection
process.)
Thus we can see that the answer, if it exists is contingent upon the random selection process. Depending on how the examiner defines “random
choice”, any or none of the answers could be considered valid. In other words, it is all in the hands of the examiner. If I am a student answering
this question I am unfortunately not privy to the examiner’s whim on this. The only thing I can conclude is that any of the available options A, B,
C or D could be considered valid by the examiner.
This leads to the second assumption – that there is one and only one option out of A, B, C or D that will be considered correct by the examiner.
And this is the normal assumption in multi-choice questions. It is how they are generally designed. We have, however, already been slammed directly
into a paradox by following natural assumptions. There is no reason to suspect the examiner will play fair. The only thing we know for sure is that
the examiner would consider either zero, one, two, three or four of the available options to be correct – by whatever perverse logic that s/he might
wish to use. I can pretty much ignore the numeric answers at this point and focus on A, B, C and D.
What is interesting here is how quickly different people are to abandon this assumption and state that there are zero correct answers. People seem
less likely to consider two, three or four correct answers.
So, it might be that the examiner considers zero of the answers to be correct. If I knew this for certain it would make answering the question
problematic. There is no way of distinguishing between answering correctly and leaving the question out. For that reason I would shy away from
leaving the answer blank. I could be wrong but it does not look like the intent of the question is to opt out.
It might be that the examiner considers one of the four answers correct. This is the default position. I would not throw out this possibility
without good reason.
It might be that the examiner considers two of the four answers correct. If this was the case then the most likely scenario is that those two answers
are A and D since they are the same. I cannot see any plausible reasoning that would render other combinations to be correct. That is not to say
such reasoning does not exist. Alternatively, the examiner might be being deliberately perverse or have faulty reasoning. But in spite of evidence
of this kind of perversity, I consider these alternatives to be less likely. If two answers are correct then they are probably A and D.
It might be that the examiner considers three answers to be correct. And again it is difficult to see the logic behind that. There might not be any.
In which case there is no real way to choose between the triples available.
It might be that the examiner considers all four answers to be correct. We have already seen that all are possible depending on the random selection
process adopted. If this is the case then it would not matter which one we chose.
Which brings us to the decision of what response to give. Much as I would like to present a series of paragraphs for a complete and justified answer,
it is stated as a multiple choice question which conventionally has me selecting from A, B, C or D. Giving a numerical response such as 25% or some
other number or shouting “elephant” while performing dance would all be responses outside of the scope of the format.
We might consider what would happen if we were actually to circle more than one of A, B, C or D. It is difficult to predict under the circumstances
how such a response might be interpreted. The best clue we have is that the question states choosing a (singular) random answer. On this basis I
would resist the temptation to select more than one option.
If there is one correct answer and I am to choose one, then the best measure I can give for its probability is 25%. This has me choosing either A or
D and I really have no way of preferring one over the other.
If there are two correct answers then, as we have seen, I should feel confident about picking either A or D.
If there are three correct answers then I confess to being bewildered as to the logic behind it. But selecting something is better than nothing and I
am likely to choose one of the three correct options.
If there are four correct answers then it really does not matter which I select – I will be correct.
So, whatever insanity the examiner adheres to, I am well served to choose either A or D as my solution. Either will do but not both.
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