We have a set of intervals so that for each two intervals
,
the disjunction is the empty set
and the union of all intervals forms a new interval from zero to
,
. Intuitively, this means that the intervals do not overlap and that there are no gaps between the intervals.
To ensure both properties mentioned above, the intervals are specified by their right hand value only. Thus, we have a set whose values define the right hand sides of all intervalls. Suppose we can define an order on the set such that
. Then the
th interval is
for
and
for
. This allows us to specify
intervals by
values only and to ensure that the intervals neither do overlap nor have gaps inbetween. Now the probability
for the
th interval is given by