I've a straight stick which has been broken into three random-length pieces.
What is the probability that the pieces can be put together to form a triangle?
If you can answer this at this point, please do.
If not, perhaps this will help: here are several methods to break the stick into the three random length pieces:
- I select, independently, and at random, two points from the points that range uniformly along the stick, then break the stick at these two points.
- I select one point, independently, and at random (again uniformly), and break the stick at this point. I then randomly (with even chances) select one of the two sticks and randomly select a point (again uniformly) along that stick, and break it at that point.
- I select one point, independently, and at random (again uniformly), and break the stick at this point. I then select the larger stick, and randomly select a point (again uniformly) along that stick, and break it at that point.
If this clarifies the problem, please show how this affects your work.
Please tell me where I’m going wrong!
Assume stick of unit length and knowledge that a piece>0.5 makes creating a triangle impossible
Case 1 is straightforward
Your ‘breaks’ are at a* and b* (where a*<b*) making pieces
a,b,c (c is the left over piece)
p(a,b both>0.5) = ¼ (making a>0.5)
p(a,b both<0.5) = ¼ (making c>0.5)
p(a<0.25 and b>0.75)= ¼ (making b>0.5)
so probability you can make a triangle is ¼
Probability clearly ½ of case 3
The longer piece, x, has an expected length of 0.75, since it is uniformly between 0.5 and 1
Consider the arbitrary breaking point, y, on the longer stick.
If y>0.5 (on a stick with expected length of 0.75) you cannot make a triangle [probability = 1/3]
If (x-y)>0.5 (on a stick with expected length of 0.75) you cannot make a triangle [probability = 1/3]
So the probability you can’t make a triangle is 2/3
The probability you can being 1/3
Intuitively I like the solution but I trust Charlie’s simulation more – I’m struggling to do my own simul’ . Can someone spot my deliberate mistake?
And Penny, this is just one of the benefits of being able to run problems through a computer
Posted by Lee
on 2003-12-08 01:39:56