Probability of forming a triangle

[winniethepooh]

Suppose you cut a stick in two random places. What is the probability that you'll be able to form a triangle out of the three pieces?

[Anurag@Gurome]

Suppose you cut a stick in two random places. What is the probability that you'll be able to form a triangle out of the three pieces?

-----x--------------M--------y-----------

Let us assume that M is the midpoint of the stick and x and y are the points where the cuts have been made.

Now a triangle is possible if and only if no part is longer than half of the length of the stick.

Now if both cuts are on the same side of the midpoint M, then no triangle is possible. The probability that both cuts are on the same side of the midpoint M is 1/2.

Now there are two possible scenarios in case both the cuts x and y are on different sides of M,

• 1. -----x--------------M--------y-----------
  x is further left in its half than y is in the right half. In this case there is no triangle possible.
  
  2. -------------x------M--------y-----------
  x is not further left in its half than y is in the right half. In this case a triangle is possible.

Now (1) and (2) have equal probability equal to 1/2.

Hence, probability that a triangle is possible = (1/2)*(1/2) = 1/4

[amit2k9]

good way to solve this.good learning.

[stuffstuff]

"Now a triangle is possible if and only if no part is longer than half of the length of the stick. "

Shouldn't it also exclude circumstances where one piece is precisely half the length of the stick?

[arnabis2good]

No side can be half the size of the stick or more.

Can we think like this?

a. Probability of the first pair of pieces to be not equal in size is 1/2
b. After the first cut is made, the probability of not cutting the smaller sized piece into 2 is 1/2

Let me know if you think this is over simplified.

[Frankenstein]

No side can be half the size of the stick or more.

Can we think like this?

a. Probability of the first pair of pieces to be not equal in size is 1/2
b. After the first cut is made, the probability of not cutting the smaller sized piece into 2 is 1/2

Let me know if you think this is over simplified.

Hi,
I don't think this is correct.
1)I don't think probability that the first pair of pieces to be not equal in size is 1/2 because, cutting it into two equal pieces is just one of the infinitely many possibilities and probability of a discrete case cannot be calculated using geometric probability for continuous space.
2)Even if the smaller sized piece is not cut into 2 pieces, we cannot assure that a triangle is formed by the 3 parts.
Example: Let the initial cuts be 4L/5, L/5
Now if 4L/5 is cut again into 3L/5 and L/5, we get the three lengths as 3L/5,L/5,L/5, which do not form a triangle.