# Probability - Q so easy that I can't seem to solve it

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?

• Where does the question state that the total number matters? Many multiple choice questions include irrelevant information ("distractors"). – whuber Jan 28 at 18:31
• 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. – fblundun Jan 28 at 19:10
• @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). – whuber Jan 28 at 19:13
• @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. – Colin Hicks Jan 28 at 20:04
• @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. – fblundun Jan 28 at 20:06

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).
• +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. – Colin Hicks Jan 28 at 19:58