Problem number 18 states, a piece of string is lying on the table. It is partially covered by three coins as seen in figure 1. Under each coin, the string is equally likely to pass over itself as seen in figure 2 or as seen in figure 3. What is the probability that the string is knotted after its ends are pulled? Here we have the basis of a knot, and we have three points where the string intersects itself. Two ways that this intersection can happen is either the green part of the string or the red part of the string are on top. So we will have three events and two options for each event, which result in 2 to the third power unique options. So there are going to be eight options for tying this knot. But how does a knot look like? Well, here we will make the most basic kind of knot that exists, which is called an overhand knot. It's kind of tough to explain how to tie such a knot, so let's tie all the possible overhand knots together. So the first step, as with any knot, is to make a loop. But here we already find a problem, because we have a choice. We can either pass the red part of the rope over the green part of the rope, or under in order to tie this knot. So let's account for both combinations. The next step is to put the string through the loop. We can only do this one way for each combination, as the alternate way would undo the loop we just made. Now for the last one, we need to make sure the string leaves the opposite way that it came in to ensure that it goes through the loop. Again our move is forced on this one. We were able to tie this knot in two ways. In the knot world, we call this a left-handed and a right-handed variant. So there were eight possible crossings, but only two of them created a valid knot. So one out of four will tie a valid knot. So the question asked us, what is the probability that the string is knotted after its ends are pulled? The answer is one in four. Letter B.

probability

string knot

overhand knot

crossing combinations

coin crossings