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!
To find the probability that y < x, given 0 < y < 1 and 0 < x < 1, we can use a graphical approach.
1. Sketching the graph:
First, draw a coordinate grid with x-axis and y-axis that range from 0 to 1. Since both y and x should be greater than 0 and less than 1, the graph will be in the square with vertices at (0,0), (1,0), (1,1), and (0,1).
2. Defining the region of interest:
Now, shade the region below the line y = x within this square. This shaded region represents the area where y < x.
3. Calculating the probability using the area method:
The probability that y < x is equal to the ratio of the shaded area to the total area. The total area is simply the area of the entire square, which is 1 (since the side length of the square is 1).
To find the shaded area, we need to find the area of the triangle formed by the line y = x, the y-axis, and the x-axis. The formula for the area of a triangle is (1/2) * base * height.
In this case, the base is 1 (since the length along the x-axis is 1), and the height is also 1 (since the length along the y-axis is 1). Therefore, the area of the shaded triangle is (1/2) * 1 * 1 = 1/2.
4. Calculating the probability:
The probability that y < x is equal to the shaded area divided by the total area: (1/2) / 1 = 1/2.
So, the probability that y < x, given 0 < y < 1 and 0 < x < 1, is 1/2.