## February 22, 2010

### Tough probability riddle

A stick of length 1 metre is divided randomly into 3 parts.
What is the probability that a triangle can be made with
those three parts?

Timi said...

100% three sticks can always make a triangle.

Secret SquÃ¯rrel said...

Not so. If one of the sticks is at least as long as the other two added together, you won't be able to form a triangle. It will just be two sticks lying against another stick.

Therefore the longest piece must be shorter than half the original length.

The probability of being able to form a triangle from the three pieces depends on how the placement of the two cuts is determined. I'm going to assume that the cuts can be made at any point along a continuous distribution and that they are independent of each other; ie the position of one cut doesn't affect the location of the other (except that they can't be in the same place since we're told that there are three pieces).

Let's assume that the first cut is made at some point. The stick can be oriented so that the shorter piece is always on the left. This way, we only have to consider half of the cases as the other will be a mirror-image of the first.

If the second cut is also made to the left of the original stick's centre, you won't be able to make a triangle Since the longest piece (right-hand part) will be at least 0.5 units long.

The probability of this occurring is obviously 1/2. So at least half the time you won't be able to make a triangle.

However, if the second cut is made to the right of the centre, you might still not be able to make a triangle.

This can occur when the first cut is made between the left end and 1/4 the way along the stick. Then, if the second cut is made less than 1/4 along from the right end, the centre piece will be at least 0.5 units long, preventing you from making a triangle. The chance of this occurring is 1/4.

1/2 + 1/4 = 3/4
So the probability that a triangle can be made is only 1/4.

Zac said...

The stick is only divided into 3 parts, it does not say its cut
so it stays as a 1 metre stick

Anonymous said...

Zac, you are completely wrong. This isnt a "trick" question. But nice try.