Originally posted by doodinthemood
You have a stick.
You snap it in a random place along the stick.
You snap it again in a random place along the stick.
The two snaps are made independent of each other. IE, the first determines nothing about the second.
What's the probability that the three pieces can make up a triangle?
As long as the length of two pieces together is more than the lenth of the third piece you will be able to make a triangle.
If you snap the stick in two halves on the first break, there will be no triangle. Fortunately, the chance of this happening is infinitely small.
Suppose you snap the stick at a/b of the total length, with b > 2a.
Now, if you make the second break in the smaller part, you will not be able to make a triangle. The chance to snap the stick in the longer part is 1 - a/b. You will always be able to make a triangle in this case.
To get a numeric value, we need to make an integral:
Suppose we get a stick of length x at the first break, with x < 1/2. The chance to make a traingle after the second break is 1-x.
2 * INTEGRAL ( 0 to 1/2) (1-x dx) should give the probability requested.
Note that the factor 2 comes from symmetry, x < 1/2 or x > 1/2.
Apologies to any mathematician/physicist if above is nonsense, it's been a while since I had to make my own integrals.
Above integral comes out to be 3/4. That I am sure of 🙂