What is the probability that a triangle can be made with those three parts?

A stick of length 1 is divided randomly into 3 parts.

What is the probability that a triangle can be made with those three parts?

Answer

The probability, that a triangle can be made by randomly dividing a stick of length 1 into 3 parts, is 25%

A triangle can be made, if and only if, sum of two sides is greater than the third side. Thus,

X1 < X2 + X3
X2 < X3 + X1
X3 < X1 + X2

Also, it is given that X1 + X2 + X3 = 1
From above equations: X1 < 1/2, X2 < 1/2, X3 < 1/2
Thus, a triangle can be formed, if all three sides are less than 1/2 and sum is 1.

Now, let's find the probability that one of X1, X2, X3 is greater than or equal to 1/2.
Note that to divide stick randomly into 3 parts, we need to choose two numbers P and Q, both are between 0 & 1 and P

Now, X1 will be greater than or equal to 1/2, if and only if both the numbers, P & Q, are greater than or equal to 1/2. Thus, probability of X1 being greater than or equal to 1/2 is = (1/2) * (1/2) = 1/4

Similarly, X3 will be greater than or equal to 1/2, if and only if both the numbers, P & Q, are less than or equal to 1/2. Thus, probability of X3 being greater than or equal to 1/2 is = (1/2) * (1/2) = 1/4

Also, probability of X2 being greater than or equal to 1/2 is = (1/2) * (1/2) = 1/4
The probability that a triangle can not be made
= (1/4) + (1/4) + (1/4)
= (3/4)
Thus, the probability that a triangle can be made

= 1 - (3/4)
= (1/4)
= 25 %

Thus, the probability that a triangle can be made by randomly dividing a stick of length 1 into 3 parts is 25%

Let's generalise the problem. What is the probability that a polygon with (N+1) sides can be made from (N+1) segments obtained by randomly dividing a stick of length l into (N+1) parts?

The probability is = 1 - (N+1)*(1/2)^N
The probability tends to 1 as N grows. Thus, it is easier to make a N-sided polygon than it is to make a triangle!!!