>I recently used this problem from the NSML contest problem database. >I can't seem to solve it directly. Any help on it would be >appreciated!
>: an 8-ft stick and a 22-ft stick are both randomly broken into >two parts. What is the probability that the longer part of the >8-ft stick is longer than the shorter part of the 22-ft stick? >
I like Gary Tupper's explanation of making the rectangle 4 x 11, but then he seems to get confused and ended up with 8 in there somewhere.
May I suggest placing coordinates on the corners of the rectangle. The coordinate (x,y) would represent x=length of short part of 22-ft stick, y=length of long part of 8-ft stick. The rectangle would have vertices at (0,4), (0,8), (11,4), and (11,8). Then draw a segment representing where these lengths would be equal, from (4,4) to (8,8).
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I think this is what Gary meant. Now any point in the trapezoid to the left has coordinates (x,y) where x<y. Area of trapezoid = 4*(4+8)/2 = 24. Area of rectangle = 4*11 = 44, so probability is 24/44.
Gary obviously did better in geometric probability than he lets on, but he needed a picture!