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Let $X$ and $Y$ be independent random variables, each of which is uniformly distributed between $0$ and $1$. Find the probability that $(X−1/2)^2+(Y−1/2)^2≤1/9$. Give at least $8$ correct digits after the decimal point.

I do not know where to start with this question. Can someone please get me on the right track?

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    $\begingroup$ Why do you need 8 rather than say 6 digits? Why do you need any digits of imprecision? P.S. There is a special tag called homework for homework questions. $\endgroup$ – wolfies Apr 29 '14 at 19:29
  • $\begingroup$ Algebraically, $(X-1/2)2+(Y-1/2)2\le 1/9$ is equivalent to $X+Y\le 19/18$, which is a considerably simpler expression. Perhaps you did not write what you intended? Maybe it should be $(X-1/2)^2+(Y-1/2)^2\le 1/9$? In either case, draw a picture of the set of $(X,Y)$ that satisfy the inequality. Then think about how you would find the probability of that set, given that $(X,Y)$ has a constant density. $\endgroup$ – whuber♦ Apr 29 '14 at 19:35
  • $\begingroup$ I did mean to have (X−1/2)^2+(Y−1/2)^2 ≤ 1/9, sorry. Algebraically, I understand that if there was only one distribution, instead of two added together, then the probability would be 2/3. How would I factor in the second distribution? $\endgroup$ – user42276 Apr 29 '14 at 19:45
  • $\begingroup$ Re the self-study tag: the exact same question has just been posed an hour ago by a different person at: math.stackexchange.com/questions/775525/… ... Except they list it as their HOMEWORK ... And you are self-studying the exact same question??? Should SE be a homework factory?? $\endgroup$ – wolfies Apr 30 '14 at 12:58
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Why don't you draw a picture? Here's a hint:

Hint

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  • $\begingroup$ Fantastic answer; it shortcuts the whole issue down to a single very clear diagram from which the algebraic answer is instantly obvious, and the approach ('draw a picture') is a very useful one on a large variety of problems. $\endgroup$ – Glen_b -Reinstate Monica Apr 29 '14 at 23:17
  • $\begingroup$ @baudolino Pretty picture - what software do you use? $\endgroup$ – wolfies Apr 30 '14 at 7:40
  • $\begingroup$ @wolfies: Geogebra. It's open source and pretty cool. $\endgroup$ – baudolino Apr 30 '14 at 14:07
  • $\begingroup$ @baudolino Cool - many thanks. I will take a look. $\endgroup$ – wolfies Apr 30 '14 at 14:42

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