Pythagorean Ant

Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a triangle with side lengths 30cm, 40cm and 50cm from some cardboard, when a gust of wind blows the triangle down into the garden.
Another gust blows a small ant straight onto this triangle. The poor ant is completely disoriented and starts to crawl straight ahead in random direction in order to get back into the grass.
Assuming that all possible positions of the ant within the triangle and all possible directions of moving on are equiprobable, what is the probability that the ant leaves the triangle along its longest side?
Give your answer rounded to 10 digits after the decimal point.

This is a very tricky probability problem, and to solve this problem, we need to understand geometrical properties of triangles and some concept of law of reflection:

1. First, observe that once the ant hits an edge of the triangle, it will follow the law of reflection (the angle at which it approaches the edge will be the same as the angle at which it leaves). This is the same behavior that a ray of light would exhibit in this situation.

2. By following a principle called “geometrical optics,” we can think of each reflection as a traverse along an extended grid of equilateral triangles (each one being the reflection of the original across one of its edges).

3. The ant moving in this extended grid will eventually move off one of the images of the 50cm side if it ever hits the 50cm side of any triangle. If it hits a 30cm or 40cm side, however, it will simply bounce off and continue moving.

4. Within this framework, we essentially want the ant to hit one of the images of the 50cm side before it hits an image of the 30cm or 40cm side.

5. For this particular triangle (a 3-4-5 right triangle), the reflected grid has 3 50cm sides (one original, two reflections), 4 40cm sides, and 5 30cm sides.

6. Therefore, the probability that the ant exits from the 50cm side (or an image of it) rather than the 30cm or 40cm side (or their images) is 3/(3+4+5) = 3/12 = 0.2500000000, when rounded to 10 digits after the decimal point.

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