The variable x takes a value between 0 and 10 with uniform probability distribution. The variable y takes a value between 0 and 20 with uniform probability distribution. The probability of the sum of variables (x + y) being greater than 20 is

The correct answer is 0.25.

Step-by-step Explanation:

1. Identify the sample space:
The variable $x$ is uniformly distributed between 0 and 10:
$0\le x\le 10$
The variable $y$ is uniformly distributed between 0 and 20:
$0\le y\le 20$
Since $x$ and $y$ are independent, their joint probability density function is uniform over a rectangular region in the 2D plane.
The total area of this rectangular region (the sample space $S$) is:
$\mathrm{Area}\left(S\right)=10\times 20=200$

2. Define the event of interest:
We want to find the probability that the sum of the variables is greater than 20:
$x+y>20$
This inequality can be rewritten as:
$y>20-x$

3. Determine the favorable region:
We need to find the area within the rectangle $[0,10]\times [0,20]$ that lies above the line $y=20-x$.
Let us find the boundary points of this region where the line intersects the boundaries of our rectangle:
- At $x=0$, $y=20-0=20$. This is the top-left corner of the rectangle: $(0,20)$.
- At $x=10$, $y=20-10=10$. This point is on the right boundary of the rectangle: $(10,10)$.
The line connects $(0,20)$ and $(10,10)$. The region where $y>20-x$ within the rectangle is a right-angled triangle with vertices at $(0,20)$, $(10,20)$, and $(10,10)$.

4. Calculate the area of the favorable region:
The base of this triangle along the top edge of the rectangle (from $x=0$ to $x=10$ at $y=20$) has a length of:
$\mathrm{base}=10-0=10$
The height of this triangle along the right edge of the rectangle (from to $y=20$ at $x=10$) has a length of:
$\mathrm{height}=20-10=10$
Thus, the area of the favorable region $A$ is:
$\mathrm{Area}\left(A\right)=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}=\frac{1}{2}\times 10\times 10=50$

5. Calculate the probability:
Since the probability distribution is uniform, the probability is the ratio of the favorable area to the total area of the sample space:
$P(x+y>20)=\frac{\mathrm{Area}\left(A\right)}{\mathrm{Area}\left(S\right)}=\frac{50}{200}=0.25$