Question

A stick of length one is broken into two pieces at a random point. What is the probability that the length of the longer piece will be at least three times the length of the shorter piece?

Answer 1

Answer: The probability that longer piece will be at least three times the lenth of the shorter piece is 0.25

Explanation:

Length of stick = 1 unit.

Let, length of shorter piece = x units

Then, minimum length of longer piece = 3x units

The minimum ratio of shorter piece to longer piece should = x/3x = 1/3

As the length of the stick is 1 and x is the length of the shorter piece of stick,

The random variable x is uniformly distributed on the interval 0 ≤ x ≤ 1,

P(0 ≤ x ≤ 1)= x/1 = x

We find the relation between ratio and the length of shorter stick as below,

Ratio (r) = shorter stick length / longer stick length = x/(1-x)

Therefore, x = r*(1-x)

Therefore, x = r-rx

Therefore, x + rx= r

Therefore, x = r/(1+r) = (1/3) / (1+1/3) =1/4

The cumulative distribution function for ratio (r) is then given by

P(r ≤ 1/3) = P((x/(1−x)) ≤ 1/3)

= P(x ≤ 1/3 / (1+1/3))

=P(x ≤ 1/4)   

=1/4      (since it is uniform distribution)

= 0.25

The probability that longer piece will be at least three times the lenth of the shorter piece is 0.25