### A problem for Pi Day

Mar. 14th, 2012 08:15 am**lederhosen**

An interesting little probability problem:

Suppose we want to divide the interval [0,1] into 100 sub-intervals (wrapping around in a ring, so 1 is 'glued' to 0). We choose to do this by generating 100 random numbers between 0 and 1 (uniform distribution, independent of one another), sorting them from smallest to largest, and using them as the borders of these intervals.

For instance, if our 100 random numbers turned out to be {0.025, 0.030, 0.038, ..., 0.971, 0.988} then our intervals would be [0.025, 0.030) and [0.030, 0.038) and ... and [0.971, 0.988) and then the 'wrapping interval' [0.988,1]+[0,0.025).

NB: it doesn't really matter for this problem, but in case you're trying to remember notation, square brackets mean the endpoint is INCLUDED, round brackets mean it's not. So the interval [0.2,0.3) would contain 0.2 but not 0.3.

Between them, these 100 sub-intervals completely cover the space between 0 and 1, with no interlap.

Two questions:

(1) What is the average length of one of these intervals?

(2) What is the average length of the interval that contains 1/pi?

Suppose we want to divide the interval [0,1] into 100 sub-intervals (wrapping around in a ring, so 1 is 'glued' to 0). We choose to do this by generating 100 random numbers between 0 and 1 (uniform distribution, independent of one another), sorting them from smallest to largest, and using them as the borders of these intervals.

For instance, if our 100 random numbers turned out to be {0.025, 0.030, 0.038, ..., 0.971, 0.988} then our intervals would be [0.025, 0.030) and [0.030, 0.038) and ... and [0.971, 0.988) and then the 'wrapping interval' [0.988,1]+[0,0.025).

NB: it doesn't really matter for this problem, but in case you're trying to remember notation, square brackets mean the endpoint is INCLUDED, round brackets mean it's not. So the interval [0.2,0.3) would contain 0.2 but not 0.3.

Between them, these 100 sub-intervals completely cover the space between 0 and 1, with no interlap.

Two questions:

(1) What is the average length of one of these intervals?

(2) What is the average length of the interval that contains 1/pi?