Spaghetti Strands
The Tramp and the Lady are down to 5 strands of spaghetti (total of 10 ends) in their bowl. They decide to pick up a random end, and tie it to another randomly selected end. They will continue to do so until no ends remain, leaving them with just loops of spaghetti. The expected number of loops in the bowl has the form , where and are positive, coprime integers. What is the value of ?
The answer is 878.
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1 solution
Suppose there are 'n' strands.
When one picks a random end, and ties it to another random end, then there are two cases:
Case 1: In this case, we form a loop and we are left with (n-1) strands.
The probability of this case:
The first end is picked at random. The probability of the second random end being from the same strand is 1/(2n-1). Net probability: 1/(2n-1)
Case 2: In this case, we are left with (n-1) 'strands' and no loops.
The probability is (2n-2)/(2n-1)
Therefore, we can see that with n strands left, when one picks a random end, and ties it to another random end, we are left with (n-1) strands. The expected number of loops formed due to this operation is 1/(2n-1).
By linearity of expectation, the required fraction is 1+1/3+1/5+1/7+1/9 or 563/315. Final answer is 878.