What is the probability that two numbers selected at random between 0 and 10 have a sum greater than or equal to 8 if the numbers are not necessarily integers?

We are tasked with finding the probability that two random numbers selected between 0 and 10 have a sum greater than or equal to 8. The numbers are not restricted to being integers, meaning they are continuous random variables.

### Step 1: Define the sample space
The sample space consists of all possible pairs of numbers \((x, y)\) where both \(x\) and \(y\) lie between 0 and 10. Geometrically, this corresponds to all points in a 10x10 square on the coordinate plane, where:
\[
0 \leq x \leq 10 \quad \text{and} \quad 0 \leq y \leq 10.
\]
The total area of this square is:
\[
\text{Area of the sample space} = 10 \times 10 = 100.
\]

### Step 2: Define the favorable region
We are interested in the region where the sum of the two numbers is greater than or equal to 8, i.e., \(x + y \geq 8\). This inequality represents a line on the coordinate plane:
\[
x + y = 8.
\]
The region where \(x + y \geq 8\) lies above and to the right of this line, bounded by \(x = 0\), \(y = 0\), \(x = 10\), and \(y = 10\).

The line \(x + y = 8\) intersects the \(x\)-axis at \(x = 8\) and the \(y\)-axis at \(y = 8\). The region of interest forms a right triangle with vertices at \((8, 0)\), \((0, 8)\), and \((10, 0)\), plus the area to the top-right of the triangle.

### Step 3: Compute the favorable area
1. **Area of the triangle**: The right triangle formed by the line \(x + y = 8\) with the \(x\)-axis and \(y\)-axis has base and height of length 8. The area of the triangle is:
   \[
   \text{Area of the triangle} = \frac{1}{2} \times 8 \times 8 = 32.
   \]
2. **Remaining rectangular area**: There is a rectangular region above the triangle with a width of \(10 - 8 = 2\) (between \(x = 8\) and \(x = 10\)) and height 10. The area of this rectangle is:
   \[
   \text{Area of the rectangle} = 2 \times 10 = 20.
   \]
   
So, the total area of the region where \(x + y \geq 8\) is:
\[
\text{Favorable area} = 32 + 20 = 52.
\]

### Step 4: Calculate the probability
The probability is the ratio of the favorable area to the total area of the sample space:
\[
P(x + y \geq 8) = \frac{52}{100} = 0.52.
\]

Thus, the probability that two randomly selected numbers between 0 and 10 have a sum greater than or equal to 8 is \(0.52\) or 52%.

Would you like more details or have any questions?

### Related Questions:
1. What is the probability that the sum of the two numbers is less than 5?
2. How would the result change if the numbers were integers instead of continuous?
3. What is the probability that two random numbers between 0 and 1 have a sum greater than or equal to 0.5?
4. What is the expected value of the sum of two random numbers between 0 and 10?
5. How do we compute probabilities for sums of random variables over other intervals?

### Tip:
When dealing with continuous random variables, probabilities often correspond to areas under curves or in geometric regions.

Explore the probability that two randomly selected numbers between 0 and 10 have a sum greater than or equal to 8. This problem uses geometric probability to compute areas of triangles and rectangles, resulting in a probability of 0.52. Ideal for high school students in Grades 10-12.

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