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On an ideal rectangular billiard table what is the probability that a ball chosen in random position will get into one of the six holes?

Edit: Assume that the billiard ball is a point, the billiard table is a rectangle $a\times b$ ($a>b$). There is no friction. Let's say the ball is in the hole if it crosses the border of the rectangle in the following ways: for angle holes, imagine an isosceles triangle with an angle in common with the table and an hypotenuse length of $l$. If the ball crosses this hypotenuse and enters the triangle, it is in the hole. For the central holes, the ball must hit the segment (length $l$, center $a/2$). The bounces are always ideal, meaning perfectly specular reflection. The ball starts moving in a random and uniformly chosen direction, as pointed out by Brian Tung.

How does it changes with uniform friction (meaning constant deceleration)?

Excuse me, details were not enough

JavierG
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    How are you modelling this? Are the holes completely within the interior of the table? – J P Feb 27 '19 at 21:05
  • Yes, I mean a standard table with holes at edges and at the half of the longest side. Say the hole lenght is k. I guess the probability I'm looking for is a function of the perimeter and of the aperture length. By simmetry considerations I'm also guessing it should be the sum of 6 equal probabilities related to each hole – JavierG Feb 27 '19 at 21:13
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    The hole length ? You mean the hole diameter ? Besides, we need information on the dimension of the table, the balls diameter, on the geometry of the holes... Pardon me but as it is, your approach is not scientific. – Jean Marie Feb 27 '19 at 21:42
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    A reasonable model is that the ball starts out on a point uniformly chosen from the surface of a table with dimensions $a$ by $b$, and the ball traverses a total distance of $c$ before coming to a stop due to friction. (I chose this because it makes the problem somewhat tractable by tesselating the plane with a bunch of $a$-by-$b$ rectangles.) If this seems reasonable to you, you should state it (or your alternative) explicitly in your problem statement. Otherwise, it is incomplete as is. – Brian Tung Feb 27 '19 at 22:25
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    Symmetry considerations suggest that the four corner pockets have equal probability, and the two side pockets have equal probabilities, but the one equal is not necessarily equal to the other equal. ;-) – Brian Tung Feb 27 '19 at 22:26
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    I forgot to mention that you should also probably stipulate that the ball starts moving in a random and uniformly chosen direction. – Brian Tung Feb 27 '19 at 22:27
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    “Unfold” the possible paths by tiling the plane with a suitable number of reflections of the table. See https://math.stackexchange.com/q/2800897/265466 and similar questions. – amd Feb 27 '19 at 22:50
  • I should have said it before, sorry :( I mean no friction, purely an ideal mathematical question – JavierG Feb 28 '19 at 01:41
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    In the frictionless case almost every trajectory covers the table densely, it is analogous to an irrational torus winding if you fold the rectangle into a torus. So eventually it will "hit" one of the triangles or segments, see also Bagchi A classification of classical billiard trajectories. – Conifold Mar 01 '19 at 11:49

1 Answers1

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Something like this? Guy hits pool balls without really aiming and it goes in 3x

heyyy
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    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Oct 27 '22 at 02:37
  • Can’t comment on original post without 50 reputation and I am being downvoted for a clarifying post? Wow.. no one wins here – heyyy Oct 28 '22 at 00:00