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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 $2y$ ?

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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.

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

enter image description here

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

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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: $$p(x > 2y) = {{1}\over{4}}$$

enter image description here

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

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