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The random variables $X$ and $Y$ are independent, each with the uniform distribution on $[−1, 1]$.

Find: $$P[\max (X,Y) >0.5]$$ Apparently there is an easy approach without integration, but I am having trouble visualizing it. Thoughts?

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possible duplicate of Expected value of max(x, y) – kaine Nov 10 '13 at 20:34

Draw the square $[-1,1]\times[-1,1]$. Shade in the region where $X$ or $Y$ is greater than $0.5$. Calculate the proportion of the shaded area to the area of the square.

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Simply note that \begin{align*} P\big(\max (X,Y) >0.5\big) &=1-P\big(\max (X,Y) \le 0.5\big)\\ &=1- P\big((X\le 0.5)\cap (Y\le 0.5)\big)\\ &=1-P\big((X\le 0.5)\big)P\big((Y\le 0.5)\big). \end{align*}

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