blades of grass

[radosr]

In a hand are six blades of grass next to each other pointing in the same direction. The tops of the blades are randomly collected into pairs that are linked together, and likewise with the bottoms of the blades. What is the probability that as a result of this operation, the six blades will be connected in a single ring?

 

[stonehead]

Despite the blades of grass model, this problem can be simplified to "Three equal ropes, randomly link the ends of them together, probability of making a single circle."

Then it's a simple probability question. 1- 1/5- 4/5*1/3 = 8/15.

(In the first step, you first randomly link two ends together. The probability of linking the ends from the same rope is 1/5. If you do that, you have failed in the first step.

If you didn't fail in the first step, then you are left with the other four ends. The probability of linking the ends from the same rope (including the linked rope in the step above) is 1/3. )

 

[C S]

Despite the blades of grass model, this problem can be simplified to "Three equal ropes, randomly link the ends of them together, probability of making a single circle."

Then it's a simple probability question. 1- 1/5- 4/5*1/3 = 8/15.

(In the first step, you first randomly link two ends together. The probability of linking the ends from the same rope is 1/5. If you do that, you have failed in the first step.

If you didn't fail in the first step, then you are left with the other four ends. The probability of linking the ends from the same rope (including the linked rope in the step above) is 1/3. )

Pretty good initial insight. For those interested, the more straightforward (I think) way to do it is notice you are counting cycle decompositions of a permutation, where you are only allowed even cycles (since a loop consists of the same number of tops as bottoms).