If 0 < y < 1 and 0 < x < 1, find the probability that

y < x. Begin by sketching te graph, and then use the area method to find the probability.

Please help me. I do not understand. Thank you in advance!

The area of the region is 1.

The line y=x divides it in half, along one diagonal.

So, the area below y=x, representing y<x is 1/2.

So, p(y<x) = (1/2)/(1) = 0.5

To find the probability that y < x, given the constraints 0 < y < 1 and 0 < x < 1, we can approach the problem by sketching a graph and then using the area method.

First, let's sketch the graph of the given constraints on a coordinate plane. Since the values of y and x lie between 0 and 1, we can draw a square with vertices (0,0), (0,1), (1,0), and (1,1) to represent the valid range of values.

Now, within this square, we need to find the region where y < x. This corresponds to the area below the line y = x within the square.

To calculate the probability using the area method, we need to find the ratio of the area where y < x to the total area of the square.

The total area of the square is 1 (since each side length is equal to 1), so we just need to find the area where y < x.

The region under the line y = x forms a right-angled triangle with vertices at (0,0), (1,1), and (1,0). The area of a triangle is given by the formula: Area = (base * height) / 2.

In this case, the base of the triangle is 1, and the height of the triangle is also 1. Substituting these values into the formula, we get: Area = (1 * 1) / 2 = 1/2.

Therefore, the probability that y < x is equal to the area where y < x divided by the total area of the square, which is 1/2 / 1 = 1/2.

To summarize:
1. Sketch the graph of the given constraints on a coordinate plane.
2. Identify the region where y < x, which corresponds to the area below the line y = x within the square.
3. Calculate the area of this region using the formula for the area of a triangle.
4. Calculate the probability by dividing the area where y < x by the total area of the square.

In this case, the probability that y < x is 1/2.