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A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction.

Dec 22, 2018

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$$\text{break the figure into 3 pieces}\\ \text{the first piece is 2x1001 white squares to the left of the shaded squares}\\ \text{the second piece is the 2 shaded squares}\\ \text{the third is the 2x1001 white squares to the right of the shaded squares}\\ \text{to pick a rectangle that does not contain the shaded squares we pick two points}\\ \text{from either the first section or the third section and these define our rectangle}\\ \text{there are }\dbinom{1001}{2} =500500\text{ ways to pick 2 points in both sections}\\ \text{so there are }2 \cdot 500500 = 1001000 \text{ ways to pick the rectangles that contain no shaded squares}$$

$$\text{There are }\dbinom{2003}{2} = 2005003 \text{ total ways to pick 2 points from the 2003}\\ \text{thus }\\ P[\text{rectangle contains no shaded square}]= \dfrac{1001000}{2005003 } = \dfrac{1000}{2003}\approx 0.4992$$

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Dec 22, 2018