Question

Divide a line segment into two parts by selecting a point at random. Find the probability that the length of the larger segment is at least three times the length of the shorter segment. Assume a uniform distribution.

Short answer

The probability that the length of the larger segment will be at least three times the length of the smaller segment given a random division is \( \frac{1}{2} \).

Step-by-step solution

Understand the problem

We want to find the probability that the length of the larger segment is at least three times the length of the shorter segment. This will be the favorable cases over the total possible cases. As it is mentioned, we work under the assumption of a uniform distribution (the point splits the line at a randomly chosen place with all places being equally likely).

Identify favourable and total cases

To deal with the uniform distribution assumption, we consider a line of length 1 (since lengths are relative we can decide the length). Then, we look for the spots on this unit line where a point would divide the line into two segments with one being three times longer than the other. This occurs in two regions of the line: between 0 and 1/4 (where the point makes the right segment three times the left one), and between 3/4 and 1 (where the point makes the left segment three times the right one). Thus, the total 'favourable' length is 1/4 + 1/4 = 1/2. For the total possible cases, any point on the line could be chosen, so the total 'possible cases' length is 1.

Calculate Probability

The probability is the favourable cases over the total possible cases, each case being a possible position for the point on the line. So, the probability is (1/2) / 1 = 1/2.