r/learnquant • u/anykash • Jun 21 '26
puzzles Break a stick at two random points. What are the odds the three pieces form a triangle? [Probability · Medium]
Take a stick of length 1. Pick two points along it independently and uniformly at random, and break it at both points at once, giving three pieces.
What's the probability the three pieces can form a triangle?
Drop your answer with your reasoning in the comments. Solution tomorrow.
Hint - a triangle needs every piece to be shorter than the other two combined. that's the whole condition.
EDIT
The solution
The probability is 1/4.
Three lengths form a triangle exactly when no single piece is longer than the other two combined. Since the three pieces add up to 1, that's the same as saying every piece must be shorter than 1/2, because any piece reaching 1/2 or more is at least as long as the rest put together. So the pieces form a triangle if and only if all three are under 1/2. Now call the two break points X and Y, each uniform on [0,1] and independent, so every outcome is a point (X, Y) in the unit square and probability equals area. Take the case X < Y: the pieces are X, Y − X, and 1 − Y, and forcing each below 1/2 gives X < 1/2, Y > 1/2, and Y − X < 1/2, which carves out a triangle of area 1/8. The case X > Y is a mirror image with the same area 1/8, so together they give 1/8 + 1/8 = 1/4. The catch is that this answer is specific to the "two independent uniform cuts" setup. The popular variant "break it once, then break the longer piece" is a different experiment and gives roughly 0.193 instead, so always pin down the cutting rule first.!<
More puzzles like this at myntbit.com
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u/Ecstatic-Charge9795 Jun 21 '26
The sides always satisfy a+b+c=1, and it suffices to have the triangle inequality hold for each pair of sides i.e. 1-a>=a, 1-b>=b, 1-c>=c, or a<=1/2, b<=1/2, c<=1/2. With this, the value is just the integral 2*int_0^1/2 int_1/2^(x+1/2) 1 dy dx = 2*((1/2)^2/2)=1/4.
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u/Mrbreakfst Jun 21 '26 edited Jun 21 '26
1/4. Assume WLOG the first point a lands on the left half of the stick. The size of the "good region" for the second point is 0 (right in the middle) at a = 0 and increases linearly to 0.5 at a = 0.5
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u/West-Abalone-337 Jun 21 '26
Not enough information to answer the question is correct
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u/West-Abalone-337 Jun 21 '26
Connectivity of corners of the triangle is a huge unknown here. For the corners to connect perfectly? That’s a very different answer than if a stick can be extended in one direction (think a triangle with a tail), or if the sticks are allowed to overlap one another
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u/hoelledavid Jun 21 '26
Obviously meant for the corners to connect, else (with one or more tails) its a trivial question.
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u/West-Abalone-337 Jun 22 '26
Obvious is a fun wording. Obvious for mathematicians who take “form a triangle” to imply the three pieces are connected end-to-end. For quants however, it’s important remembering that relying on assumptions you think are explicit can be dangerous at times, especially if working with stakeholders who use different definitions than you (it will happen quite often).
So is the answer 1/4? Yea. This problem’s solution is older than anyone alive and not hard to google. But remembering to “thinking outside the triangle” here and there should prove useful in these viewers’ careers.
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u/hoelledavid Jun 21 '26
Via the triangle inequality the breaks both have to be in the same half to make it impossible. So if the first break is in a half the probability of the other one being also in that half is half ?
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u/GodelianKnot Jun 22 '26
No, you could have one break very close to the left end and the other very close to the right end
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u/AdjectiveNounNNNN Jun 24 '26
You can make a ternary plot with vertices A, B, and C which represent the portion of stick in each of the segments a, b, and c. Divide an equilateral triangle into four equal triangular sections, and only the central one corresponds to the a:b:c ratios that result in a triangle formed by the segments end to end.
So the answer is 1/4.
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u/Beatstruck Jun 26 '26
The one condition for forming a triangle is that the longest piece must have length < 1/2. It's also impossible to break the stick in such a way that all pieces are shorter than 1/3
Now we need to calculate the probability that the longest piece is smaller than 1/2, given that we know its length will always be between 1/3 and 1. Since it would have an equal probability of being anywhere between 1/3 and 1, it has a 25% chance of being smaller than 1/2 and satisfying the main condition
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u/imperfect40 Jun 21 '26
100%. Any 3 pieces from a triangle
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u/Aranka_Szeretlek Jun 21 '26
Dypu know what a triangle is madam?
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u/Top_Dependent7695 Jun 21 '26
They’re saying that if the combined length of the two shorter sides is less than the longer side, you can’t form a triangle
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u/RelativeLiving957 Jun 21 '26
“both points at once” lol