# Probability: 2 random numbers such that $x > 2y$

This is probably an easy one, but I'm new to probability and really don't know how to solve this question.

Select two numbers randomly from interval $[0, 1]$ call them $x$ and $y$, what is the probability that $x$, (the first number) is greater that $2y$ ?

Hint: Draw the part $T$ of the square $[0,1]\times[0,1]$ defined by the inequation $x\gt2y$. (Sub-hint: $T$ is a triangle.) Measure the area of $T$. Conclude.

• Just for my understanding, what do you mean with T? What's the definition? – Hans Mar 20 '13 at 7:20
• @Hans see my picture. $x>2y$ is the portion of the rectangle with $(x,y)$ coordinates below the line $y=\frac{1}{2}x$ – Rustyn Mar 20 '13 at 7:59
• What's the definition? Please read: the part T of the square [0,1]×[0,1] defined by the inequation x>2y. – Did Mar 20 '13 at 8:10
• @RustynYazdanpour Thanks for the picture. – Did Mar 20 '13 at 8:11
• @Did No problemo – Rustyn Mar 20 '13 at 8:13

$T$ is the shaded part of this picture:

$x>2y$ is the portion of the rectangle with $(x,y)$ coordinates below the line $y=\frac{1}{2}x$

If $y = t$, then valid $x$ points are located above the ${x}\gt{2t}$ (see yellow area in image). Probability is the contents of the yellow triangle(${{{{1}\over{2}}\cdot{1}}\over{2}}$) divided by the contents of the entire square (${1}\cdot{1}$).
So the value being queried: $$p(x > 2y) = {{1}\over{4}}$$

• Thx, this really helps me to understand the situation :) – Hans Mar 23 '13 at 7:52