Question

Two numbers are selected at random from the interval \((0,1)\). If these values are uniformly and independently distributed, by cutting the interval at these numbers compute the probability that the three resulting line segments form a triangle.

Short answer

The probability that the three resulting line segments form a triangle is 0.5

Step-by-step solution

Identify the conditions

First, suppose that the two numbers chosen are \(x\) and \(y\) with \(0 < x < y < 1 \). The three segments have lengths \(x, y - x, 1 - y\) . Now, since a triangle is formed if the sum of lengths of any two sides exceeds the length of the third side, we thus have our three conditions: \(x + (y-x) > 1-y, x + (1-y) > y - x, y - x + (1-y) > x\)

Simplify the conditions

Now let's simplify these conditions. They respectively become: \(y < 0.5, x > 0.5, x < 0.5\) . Upon observing, we can see that the second and third conditions contradict each other. Therefore, the possible condition for triangle formation is \(y < 0.5\).

Calculate the probability

Since \(y\) is selected uniformly at random from \((0,1)\), the probability that \(y < 0.5\) is 0.5.

Key concepts

Uniform Distribution

When we talk about numbers being chosen 'at random' from a certain interval, such as \(0,1\), a common assumption in probability theory is that we're dealing with a uniform distribution. Imagine a line segment from 0 to 1. Under uniform distribution, every point on this segment is equally likely to be chosen when we pick a number at random.

In the context of our triangle formation problem, this means that any point within the interval \(0,1\) has the same chance of being selected as the cutting point for our segments. This equal likelihood is critical for calculating probabilities because it simplifies our computations - we don't have to worry about some numbers being more likely than others. With this in mind, if an event (like choosing a number less than 0.5) covers half of our interval, then intuitively, the probability of this event is 0.5, which is just the proportion of the interval that satisfies our condition.

Triangle Inequality

The triangle inequality is a fundamental principle used to determine if three line segments can form a triangle. This principle states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

In our exercise, after cutting our line segment into three pieces with lengths \(x, y-x, 1-y\), we need to check their suitability to form a triangle using the triangle inequality: The sum of the lengths of any two segments must exceed the third's length. For a deeper understanding, it's important to visualize this: if you have two sticks, no matter how you place them, you can't touch the ends without a third stick that is longer than the gap between the other two ends. If your third stick is too short, there's no way to close the triangle. This mental visualization helps to grasp the concept even before delving into the mathematical inequalities that express the same idea.

Geometric Probability

Geometric probability involves using geometric shapes and measurements to calculate probabilities, often visualized on graphs or diagrams. In our problem, we visualize geometric probability through the unit interval and the possible points at which it can be cut to form a triangle.

By interpreting our conditions with triangle inequality, we could theoretically plot these conditions on a two-dimensional graph, where the x-axis is one cut and the y-axis is the other. This method will allow students to 'see' the probabilities for any region that satisfies the triangle formation conditions. Essentially, geometric probability often brings an abstract concept into a graphical representation that can be more easily understood and worked with. In this case, the probability that the three segments form a triangle is the proportion of the area under the curve that satisfies the triangle inequality, which turns out to be a relatively simple calculation because of the uniform distribution of our variables.