+52 A piece of cord is laid out along the path shown on the left. At each of the six points where the path intersects itself, by random choice the cord can either pass over or under where it has already been laid. One possible combination of choices is shown on the right. What is the probability that when the ends of the cord at points and are pulled taut the cord will form at least one knot? Express your answer as a common fraction.
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thanks!
+1459 Consider the path as a series of 6 decisions (over or under) to be made.
There are 2^6=64 total possible choices, and exactly half of them will result in a knot being formed when the ends are pulled taut.
Therefore, the probability is 1/2.
