Not Knot?

Probability Level 2

There is a piece of string lying on the floor in the position shown above. We are too far away to tell which section of the string lies above or below at each of the crossovers at X, Y, and Z.

If it is equally likely that either section lies on the top of each crossover, what is the probability that the string is actually knotted?

\frac{1}{4}

\frac{1}{2}

\frac{3}{4}

\frac{1}{3}

1 solution

Brian Charlesworth
Mar 11, 2015

We'll track the string as we move from point $X$ to $Y$ to $Z$ such that the section of string we are following is either over, $O$ , or under, $U$ , the other section of the string.

There are then $8$ possible configurations, of which only $2$ , namely the sequences $O,U,O$ and $U,O,U$ , result in knots. That is, the sequence that alternate are the only ones that can result in a knot. (This is a hands-on problem; get a piece of string and work through the $8$ possible sequences to convince yourself that these two sequences are the only ones that produce a knot.)

The desired probability is then $\dfrac{2}{8} = \boxed{\dfrac{1}{4}}.$

Nice problem, Kevin. It was hard enough going through 8 options; I'd go crazy if I had to solve the general problem involving $N$ points of overlap by counting; I think I'd have to read up on knot theory before tackling the general case. :)

Brian Charlesworth - 6 years, 3 months ago

You can cut your work in half by assuming that WLOG, X has the top-left to bottom-right string as "over". :)

A good place to get started is the knots wiki , which I see has been added as the skill :)

Calvin Lin Staff - 6 years, 3 months ago

I've generally avoided knot theory in the past, (it makes my head spin), but I'll give it another go thanks to this great wiki. :)

Brian Charlesworth - 6 years, 3 months ago

Thanks. I was playing around with some strings and dropped one on the floor. That led to this question (after some adjusting).

Chung Kevin - 6 years, 3 months ago

@Brian Charlesworth I have always observed that almost every problem on probability is linked up with some other theory (like combinatorics and knots here) can we have problems that bring out the true nature of probability rather than those that require it only at the end :p

what are your views on this sir?

Abhinav Raichur - 6 years, 2 months ago

Yes, your observation seems accurate; probability rarely plays the "starring role" in problems. Usually with such problems the difficult part is interpreting a scenario so that we can make accurate use of probability theory; the calculations themselves then usually end up being straightforward. I've just posted a question here in which I was trying to put probability at the forefront, but again it was more about the interpretation of the scenario than the actual probability calculations. I'll try and think of another problem that is more directly about probability theory.

Brian Charlesworth - 6 years, 2 months ago

Not Knot?

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