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Problem of the Day #75: Running Away from a Monster June 2, 2011

Posted by Billy in : potd , trackback

Albert is at the origin of the coordinate plane. Unfortunately, so is a very scary monster. Albert wants to maximize his distance from the monster. However, because he cannot think straight (due to the scariness of the monster), he can only move by randomly picking a direction parallel to the $x$- or $y$-axis and taking a step that direction. After each step Albert’s step length decreases by half. For example. Albert’s first move could be to any of $(1,0)$, $(0,1)$, $(-1,0)$, or $(0,-1)$, and if he decided to move to $(1,0)$, his next move could be to any of $(\frac{3}{2},0)$, $(1,\frac{1}{2})$, $(\frac{1}{2},0)$, or $(1,-\frac{1}{2})$. Albert continues this ad infinitum. If he ends on the point $(x,y)$, what is the probability that $\frac{1}{2}\leq|x|\leq\frac{3}{2}$ and $\frac{1}{2}\leq|y|\leq\frac{3}{2}$?

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