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Twenty students take an exam. Student A scored five out of ten and Student B scored eight out of ten. Every score (1 to 10) is equally likely. What is the chance of a random person out of the people that took the exam scoring higher than Student A, but lower than Student B?

So this is from a very basic quiz, no distributions or anything, just frequencies so far. It is a multiple choice question and the possible values are a) 0.8, b) 0.4, c) 0.2, d) 0.22

The random person would have to score either 6 or 7 and there is a 0.2 chance of falling into that range if scores are equally likely. What throws me off is: why does it matter how many people took the exam? The probability of score 6 or 7 must be 0.2 for the whole population of the world. Or is it that Student A and B are already out of the equation, so we are only counting with 18? But how does that matter? I must be having a mental block - can someone help, pls?

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  • $\begingroup$ Where does the question state that the total number matters? Many multiple choice questions include irrelevant information ("distractors"). $\endgroup$
    – whuber ♦
    Commented Jan 28, 2021 at 18:31
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    $\begingroup$ I think this question is poorly worded. It makes it sound like A and B were among the 20 students taking the exam - but then a randomly chosen student who took the exam would have a 2/20 chance of being A or B, in which case they definitely wouldn't have scored 6 or 7. But 0.18 isn't among the allowed answers. $\endgroup$
    – fblundun
    Commented Jan 28, 2021 at 19:10
  • $\begingroup$ @fblundun I don't have any difficulties with the wording: taken as written, it is internally consistent and unambiguous. It doesn't matter whether A and B were among those students, either (but it would be difficult to make sense of the question if they weren't). $\endgroup$
    – whuber ♦
    Commented Jan 28, 2021 at 19:13
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    $\begingroup$ @whuber I disagree, I think it’s a totally valid interpretation to think that they want the probability of some person out of the remaining 18 to score that 6 or 7 score with $1-0.8^{18}=0.982$ chance of occurring. This is not a law or language class and the ask should be quite clear from the question. Irrelevant information is one thing, but confusing wording imho destroys the very integrity of testing a students ability on the subject at hand. $\endgroup$
    – Colin Hicks
    Commented Jan 28, 2021 at 20:04
  • $\begingroup$ @whuber I think the question can easily be interpreted as "Twenty students including A and B took the exam. Given that A scored 5 and B scored 8, what's the probability that a randomly picked student (possibly A or B) scored 6 or 7?" The answer would then be 0.18. $\endgroup$
    – fblundun
    Commented Jan 28, 2021 at 20:06

1 Answer 1

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A random student will score $X$ points where $P(X = 6) = P(X=7) = 1/10.$ So $P(5 < X < 8) = 2/10 = 0.2.$ Answer (c).

I does not matter how many people took the exam. The question is about one randomly chosen student.

Note: There are authors who think it is ever so clever to clutter questions with irrelevant information in order to make sure you pay attention only to the facts that matter. Is this really a clever way to write textbooks? The answer depends on whether the aim is to teach reading or probability.

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    $\begingroup$ +1 for the comments in the end. I also dislike that someone could interpret this as finding the probability that any of the other 18 students scored a 6 or 7, which is $1-0.8^18=0.982$, only to get zero credit because of tricky writing and a somewhat ambiguous question. Thankfully this is multiple choice and that answer isn’t there, but it still is quite frustrating that they would do this. $\endgroup$
    – Colin Hicks
    Commented Jan 28, 2021 at 19:58
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    $\begingroup$ @ColinHicks: While we're at it: Under what realistic model of student behavior would scores 1 through 10 be equally likely? $\endgroup$
    – BruceET
    Commented Jan 28, 2021 at 21:01
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    $\begingroup$ @Colin and Bruce: Neither of you really needs reminding, but such lack of realism and the incapability of the OP to respond to inquiries for clarification are the salient reasons why we treat self-study questions in such special ways. $\endgroup$
    – whuber ♦
    Commented Jan 28, 2021 at 22:37
  • $\begingroup$ Thank you to everyone who responded. @whuber, I don't understand your "incapability of the OP to respond" comment: I posted the question yesterday evening, it is now the following morning - please allow me some time. The question was an online quiz, as indicated in my original post, and I do not have any more information, or ways to clarify it more. I totally agree with the comments that the "any score is equally likely" part is completely dumb - to me it says that no matter what you achieve in the exam, your score will be defined by a lottery. $\endgroup$
    – Reader 123
    Commented Jan 29, 2021 at 10:34
  • $\begingroup$ Reader: Your comment elucidates my meaning: " I do not have any more information, or ways to clarify it more." That's exactly the difficulty we face with all textbook, exam, and other self-study questions: they come out of the blue, often unexplained, and usually there is nowhere one can go to learn about their motivation or context. The problem isn't with the student, but with the very nature of the question. $\endgroup$
    – whuber ♦
    Commented Jan 29, 2021 at 15:39

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