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Two numbers $u_1$ and $u_2$ are selected uniformly randomly and independently in the interval $[0,1]$. What is the probability that $u_1$ is in $[0, 0.5]$ and $u_2$ is in $[0.5+u_1, 1]$? In other words, we are computing the probability of $u_2 - u_1 > 0.5$ assuming that $u_2 > u_1$.

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There is some ambiguity. Do you want the probability that $u_2-u_1\gt 0.5$ given that $u_2\gt u_1$? Or as in the title do you want the proability that $u_1$ is in $[0,0.5]$ and $u_2$ is in $\dots$? –  André Nicolas Aug 28 '13 at 4:23
    
Considering u2 > u1 as the fact. –  q0987 Aug 28 '13 at 14:23
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3 Answers

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If I remember correctly, this is equivalent to the following. Imagine the probability space is the unit square. The area of possibility is only the area bounded by the line $y = x + 0.5$, the line $y=1$, and the line $x=0$. Any point inside that triangle has its $y$ value at least 0.5 greater than or equal the $x$ value. This triangle is $\frac{1}{8}$ of the unit square, so the probability should be $12.5\%$.

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Hint: Assuming that $u_1$ has been chosen such that $0\le u_1\le 0.5$, what is the probability that $u_2$ is chosen so that $0.5+u_1\le u_2\le 1$? What is the probability that $0\le u_1\le 0.5$? How does conditional probability work?

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Draw a square $[0,1] \times [0,1]$ Let the horizontal axis be $u1$ and the vertical axis be $u2$. This represents the sample space. You are trying to find the region that satisfies the constraints and evaluate its area, which will be the desired probability. The first one $0 \le u1 \le 0.05$ is not hard to see. What part of that area satisfies the second condition?

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