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On table of $30cm$ x $70cm$ devided on equal right isoscales triangles where sides equal $10cm$ falls coin of radius $1cm$. What is the probability of that coin doesn't intersect any side of the triangle?

I have a big problem with determine the condition when the coin will touch the side of triangle.

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up vote 1 down vote accepted

If I understand correctly - the table is tiled with 10x10 isosceles triangles. If so, you can consider the intersections with the horizontal, vertical and (two sets of) diagonal lines independently.

So, for example, there will be 4 lines parallel with the 70 cm side (at 0, 10, 20, and 30 cm) - can you calculate the probability that the coil will not be touching one of these?

If so, repeat for each line set. Then your solution will be product of these four probabilities.

For example - for the 4 horizontal lines - if the coil is within 1 cm of the line it will overlap, so you have 1 cm unsafe for each line on the edge and 2 cm unsafe for each of the two middle lines. Total is 6 'unsafe' and 24 'safe' out of 30 cm. So probability it wont touch a horizontal line is 24/30. You can do the same easily for the vertical lines. The diagonal ones will take a little thinking about but if you think about the boundary conditions along the edges (and the fact that a 'short' diagonal in the corner is complimented by a 'long' diagonal in the opposite corner) you should eb able to get there.

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ok, now I understand how it works, but could you give me hint how to calculate the probability in your example? – Mark Dec 1 '13 at 22:54

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