You have a drawer with n shoelaces in it. You reach into the drawer and select two lace ends at random, and tie them together. Repeat this until there are no more untied lace ends left.

If the two ends are from the same lace, or from the same group of laces already tied at their ends, it will form a loop, and therefore no longer have any free ends.

Continue this until all laces are parts of loops, of either 1 lace, 2 laces, etc.

When you're finished, a certain number of loops will have been formed in this fashion, consisting of one or more laces.

What is the expected number of loops?

The required expected number of loops is:

 1 + 3^-1 + 5^-1 + .....+ (2n-3)^-1 + (2n-1)^-1

Edited on October 5, 2008, 2:32 am

Posted by K Sengupta on 2008-10-05 02:31:27