Hard math question?

Question

A bag contains n pieces of string of equal length. Two ends of string are chosen at random and tied together. Two other ends are selected and tied together. If this process continues until there are no more loose ends, what is the probability that all n pieces of string form one big loop?

Answer 1

Tracy,

 

I'll get you started.

 

One way to do this is to figure out the number of ways of tying all the ends with no restrictions, let's call that N and the number of ways it can be done resulting in one loop, let's call that N_A.

 

The probability will be N_A/N since the possibilities are equally likely.

 

I'll start with the N_A. With n strings you have 2n "ends". There are 2n ways of picking the first end and (2n-2) ways of picking the second end (since you don't want to pick as the second end the other end of the first string you picked.)

 

Therefore there are (2n)(2n-2) ways of picking the first two ends.

 

Now the third pick can be done (2n-2) ways since two ends have been tied together and are gone; then the next pick can be done (2n-4) ways since three ends are used up and you don't want to pick  the other end of the string just selected. So the next "tie" can be done (2n-2)(2n-4) ways.

 

And so on. Each of the products would the be multiplied together to get N_A.

 

Next of course you need N to get the probability. Can you do that part?

 

 

Any questions?