Post #2153521

2025-09-02 14:00 UTC

A triangle has edges of length 2, 3 and 4. Pick two points P and Q at random in the interior of the triangle, and draw the line L that contains both P and Q. Which two edges of the triangle is the line L most likely to intersect?

Replies (7)

@gregeganSF@mathstodon.xyz 2025-09-03 14:00

All pairs are equally likely! Congratulations to the many people who picked this. Why are the probabilities equal? Because you can perform a linear transformation on *any* nondegenerate triangle that turns it into an equilateral triangle, where the probabilities are obviously equal. But such a transformation preserves the ratio between the areas of any two regions, and hence the probabilities.

Open ##3179036
@tarheel@mstdn.io 2025-09-02 14:03

@gregeganSF@mathstodon.xyz I always pick the naive answer. Let's see how this turns out.

Open ##3179037
@gooba42@mastodon.social 2025-09-02 14:41

@gregeganSF@mathstodon.xyz Ouch. I picked a rational but wrong answer based on a flawed heuristic but it was an interesting question at least.

Open ##3179038
@cascheranno@hachyderm.io 2025-09-02 14:51

@gregeganSF@mathstodon.xyz area nearest 4 is largest, then area nearest 3. Statistical likelihood of picking points in these two are higher than picking one in area nearest two. Not sure about extrapolating from there to likelihood of the line’s directions. I know one ray has to cross 3 or 4 for each pair simply because any line will cross two edges. Hmm, from one point taken randomly, for any second point, the odds of crossing an edge are going to be proportional to the number of points in that direction, which again weighs toward the higher areas near 3 and 4. More points to be selected at random means higher odds of selection . That third paragraph makes me think an algorithm picking pairs and determining their crossings would favor 3 and 4. Numeric models approach algebraic at large n, so 3 & 4.

Open ##3179039
@marmarta@chaos.social 2025-09-02 15:01

@gregeganSF@mathstodon.xyz I think I picked the wrong answer, but now you made me try to remember my worst math disciple, geometry (thank you)

Open ##3179041
@Josh_Gallagher@techhub.social 2025-09-02 15:28

@gregeganSF@mathstodon.xyz I picked the wrong answer, but as a result learned about affine maps, so I feel like a winner.

Open ##3179042
@Ianagol@mathstodon.xyz 2025-09-03 01:39

@gregeganSF@mathstodon.xyz Hint: the answer is invariant under affine transformation.

Open ##3179043