Answer to Question #147517 in Statistics and Probability for Amir

Question #147517
A rectangle with height and width equal to 3 and 25 respectively, is drawn on a checkered paper. Bazil paints a random horizontal 1×2 rectangle, and Peter paints a random vertical 2×1rectangle (each rectangle consists of 2 sells). Find the probability that at least one of the cells is painted twice. Express the answer in percent, and round to the nearest integer.
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Expert's answer
2020-12-01T03:28:22-0500

As we can see, it is possible to paint 25 different horizontal 1×21\times2 rectangles. In a similar way we conclude that on rectangle 3×193\times 19 it is possible to paint 324=723 \cdot 24 =72 horizontal 1×21\times2 rectangles and 225=502 \cdot 25=50 different vertical 2×12\times1 rectangles. We have 36003600 possible combinations in total.

On 2×22\times2 square it is possible to paint 4 different intersections of vertical and horizontal rectangles. We can consider 224=482 \cdot 24=48 different 2×22\times2 squares on the rectangle. Thus, we obtain 484=19248 \cdot 4=192 different combination, when two rectangles intersect each other. I.e., there will be cells painted twice.

Thus, the probability that one cell is painted twice is: p=1923600=0.05=5%p=\frac{192}{3600}=0.05 = 5\%

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