MHB Probability of |x-y|>=6 for Two Numbers in Range [0,10]

[anemone]

Here is this week's POTW:

-----

Two real numbers $x$ and $y$ are chosen in the range $[0,\,10]$.

What is the probability that $|x-y|\ge 6$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution::)
1. MarkFL
2. Opalg

Solution from MarkFL:

Spoiler

Let's let $(x,y)$ be a point in the plane. Now, we know we must satisfy:

$$x-y\ge6\implies y\le x-6$$

$$y-x\ge6\implies y\ge x+6$$

We are also given that the point $(x,y)$ is on the square having the diagonal vertices $(0,0)-(10,10)$.

Hence, the shaded areas represent the points satisfying the given conditions:

\begin{tikzpicture}[>=stealth, xscale=1, yscale=1, font=\large]
\foreach \I in {0,1,2,3,4,5,6,7,8,9,10} {%
\draw (\I,.1) -- (\I,-.1) node[below] {$\I$};%
}
\foreach \I in {0,1,2,3,4,5,6,7,8,9,10} {%
\draw (.1,\I) -- (-.1,\I) node

{$\I$};%
}
\draw[->] (-0.5,0) -- (10.5,0) node

{$x$};
\draw[->] (0,-0.5) -- (0,10.5) node[above] {$y$};
\draw[domain=6:10, smooth, variable=\x, ultra thick, blue] plot ({\x},{(\x)-6}) node

{$y=x-6$};
\fill [green, domain=6:10, variable=\x]
(6, 0)
-- plot ({\x}, {\x-6})
-- (10, 0)
-- cycle;

\draw[domain=0:4, smooth, variable=\x, ultra thick, blue] plot ({\x},{(\x)+6}) node

{$y=x+6$};
\fill [green, domain=0:4, variable=\x]
(0, 10)
-- plot ({\x}, {\x+6})
-- (4, 10)
-- cycle;
\draw[step=1cm,gray,very thin] (0,0) grid (10,10);
\end{tikzpicture}

Now, the required probability is the ratio of the shaded areas to the area of the entire 10X10 grid, hence:

$$P(X)=\frac{4^2}{10^2}=\left(\frac{2}{5}\right)^2=\frac{4}{25}$$​

​