Official solution:
SOLUTION: The key here is to recognize that you start with 100 spaghetti-ends and each tying operation reduces the number of spaghetti-ends by two. The operation results in a new loop only if the first end you pick up is matched to the other end of the same strand. At step k, there are 100 - 2(k-1) ends, thus 100 - 2(k-1) - 1 = 101 - 2k other ends you could tie any given end to. In other words, there are 101 - 2k choices of which only one makes a loop; the others just make two strands into one longer one.
It follows that the probability of forming a loop at step k is 1/(101 - 2k). If X(k) is the number of loops (1 or 0) formed at step k, then the expected (that is, average) value of X(k) is 1/(101 - 2k). Since the final number of loops is X(1) + . . . + X(50), we conclude that on the average, the final loop count is
1/99 + 1/97 + 1/95 + . . . + 1/3 + 1/1
which is about 2.93777485; less than three loops!
