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.