A simple probability theory question, but the answer may not be simple.

Hi, I have the following question.

Suppose I draw a measure of 0<m<1 points uniformly without replacement from [0,1]. What is the probability that less than or equal to a measure of 0<k<m<1 points are drawn from the interval [i,1]? WLOG, suppose k<1-i.

An example: a measure of 1/3 points are drawn uniformly without replacement from [0,1]. What is the probability that less than or equal to a measure of 1/4 points are drawn from [1/2,1]?

I think the probability that an exact measure of 1/4 points are drawn from [1/2,1] is 0, so I am asking the CDF in the question. Hope I am correct.

Thank you very much.

Clarify the first sentence:

Clarify the first sentence:

I mean draw the points randomly from the interval [0, 1], with each remaining point being drawn with equal probability.

Thanks.

kaoyanone said:

Suppose I draw a measure of 0<m<1 points uniformly without replacement from [0,1]. What is the probability that less than or equal to a measure of 0<k<m<1 points are drawn from the interval [i,1]? WLOG, suppose k<1-i.

Click to expand...

@kaoyanone, are you using some sort of translation service to post your questions?
I ask because the vocabulary is like none I have ever seen in probability.
What does "a measure of 0<m<1 points" mean??

Your example does not help because the concept is not clear.

I think that this is a standard question about two points chosen at random from [0,1].
But it is not clear what else the question is asking about the points.

Please try again.

To "draw" some things, whether numbers or points, requires that the number of things be a positive integer so "0< m< 1" is impossible.

Thanks for reply.

I should make the question more precise: I want to draw one set of Lebesgue measure 0<m<1 uniformly from all sets of Lebesgue measure m in [0,1].

I just got known from other math forums that the set of all sets of Lebesgue measure m in [0,1] is not measurable, so there is no the ordinary probability measure on that set. So this question itself is well-defined.

Thanks again.

HallsofIvy said:

To "draw" some things, whether numbers or points, requires that the number of things be a positive integer so "0< m< 1" is impossible.

Click to expand...

pka said:

@kaoyanone, are you using some sort of translation service to post your questions?
I ask because the vocabulary is like none I have ever seen in probability.
What does "a measure of 0<m<1 points" mean??

Your example does not help because the concept is not clear.

I think that this is a standard question about two points chosen at random from [0,1].
But it is not clear what else the question is asking about the points.

Please try again.

Click to expand...

So this question itself is not well-defined.

So this question itself is not well-defined.

kaoyanone said:

Thanks for reply.

I should make the question more precise: I want to draw one set of Lebesgue measure 0<m<1 uniformly from all sets of Lebesgue measure m in [0,1].

I just got known from other math forums that the set of all sets of Lebesgue measure m in [0,1] is not measurable, so there is no the ordinary probability measure on that set. So this question itself is well-defined.

Thanks again.

Click to expand...