sticks and triangles

if we break a stick at any two points, what is the probability that the three pieces form a triangle?

try the problem out for yourself if you want! scroll down to see the explanation :D

spoiler blocker :)

hi!!!! ^-^

try dragging the points on the number line and on the plot to observe the behavior of the triangle on the bottom. this is how you will be able to tell when the break points do and dont form a triangle.

when interpreting the red area as the region of all possible break point positions, and the green area as the region of all possible break points that help to form a triangle, our answer becomes 25%!

let's walk through some of the reasoning...

question: where did the 2d plot come from?

answer: we use a 2d plot so that it is easier to reason about the probability. if we have the first point be the x-axis, and have the second point be the y-axis, our total area becomes a square since both of our points move independently. once we also begin to plot our restrictions, then we know our specific "desired" region, and finally can use our eyes to see what ratio of the "desired" region takes up the total region. in our specific case, the restrictions applied are:

x refers to the first point (or our white point in the visual)
y refers to the second point (or the pink point in the visual)
y > x, as we assume the second point is larger than the first. this is valid because no matter how we label the break points, it is true that one will be farther along the stick than the other.
x < 1/2, because if one piece is more than half the length of the stick, then no matter how we split our second piece, the resulting pieces wont be able to create a full triangle.
y > 1/2, for the same reasoning as above
y - x < 1/2, for the same reasoning as above

question: what are the circles for in the triangle visualization?

answer: i was thinking about not including them but they were too cool not to have. youll notice that the radius of each circle is the length of the first and third line segment made as a result of both breaks. youll also notice that the third point of the triangle is the intersection of both these circles. this makes a lot more sense once you think about the segments rotating around the break points like hinges. if you can turn both doors and they collide with each other, then you have "valid" break points. if you can make full revolutions on both doors without them touching, then you have "invalid" break points. (this doesnt explain the cases where i.e. the first segment is MUCH bigger than the third but the door analogy is the best i have okay)

question: why aren't there more questions?

answer: because it's hard to think in the mind of a person who's reading this and i haven't been asked more than the current list yet. if you have any questions, feel free to ask becuase i will likely add it here!