A stick is broken into two at random, then the longer half is broken again into two pieces at random. What is the probability that the three pieces make a triangle?
I've been stumped at this question for a while now, and cannot seem to find a convincing answer on the internet. The answers that I find are usually 0. 386 (which is 2ln2 - 1), but I have worked the answer out to be 0.27865.
I let X be the event of the position of the first cut of a length of unit 1. Then X ~ Unif(0,1).
Then I let Y be the event of the position of the second cut. Finding the PDF of Y is slightly more tricky.
Y is Unif (a/(1-x)) for 0 < x < 0.5 and Unif (a/x) for 0.5 < x < 1
where the normalising constant a, I worked out to be 0.5/ln2.
Now the answers I have seen did not work out this normalising constant, and you will get 0.386 as the probability to create a triangle, but with the normalising constant, I get an answer of 0.27865...
Now, as with these probability questions, you can always make an experiment, and estimate a probability. I was too lazy to do this and found a simulation here:
I found the simulation to support the 0.38 answer... but then again, I don't know how they created the simulation.
What have I done wrong here? (Or am I right?) Thanks so much!