What is the probability that if you cut a straight stick at 2 arbitrary places, you can make a triangle?

Question

The triangle is made with the pieces of the stick.

Answer 1

Addition to Answer:

Thanks to Bigsky_52 and Roger_S for digging into the problem further and finding the Right Answer. Looking at my own numerical solution, the criterion I used to determine if a combination could make a triangle or not was incorrect. To get the correct result, replace the following line of code:

130 IF A + B > C THEN TRIANGLES = TRIANGLES + 1

with:

130 IF (A
The original line forgets that any of ‘a’, ‘b’, or ‘c’ can be the “long side”, not just ‘c’. Thus the incorrect answer of 75% comes out. The new line does essentially the same test, but does so by making sure no piece is longer than half the total stick length, and therefore considers all variations and avoids the “extra” incorrect counts. If I had looked at the original ‘a’, ‘b’, and ‘c’ data more carefully, the bogus counts would have stood out.

Using the corrected program, the results are:

d = 1 … TOTAL = 498,501 … PROBABILITY = 24.92493%
d = 2 … TOTAL = 124,251 … PROBABILITY = 24.84970%
d = 4 … TOTAL = 30,876 … PROBABILITY = 24.69880%
d = 5 … TOTAL = 19,701 … PROBABILITY = 24.62312%
d = 8 … TOTAL = 7,626 … PROBABILITY = 25.60976%
d = 10 … TOTAL = 4,851 … PROBABILITY = 24.24243%
d = 20 … TOTAL = 1,176 … PROBABILITY = 23.46939%
d = 25 … TOTAL = 741 … PROBABILITY = 23.07692%
d = 40 … TOTAL = 276 … PROBABILITY = 28.26087%
d = 50 … TOTAL = 171 … PROBABILITY = 21.05263%
d = 100 … TOTAL = 36 … PROBABILITY = 16.66667%
d = 125 … TOTAL = 21 … PROBABILITY = 14.28572%
d = 200 … TOTAL = 6 … PROBABILITY = 50.00000%
d = 250 … TOTAL = 3 … PROBABILITY = 0.00000%

As ‘d’ gets smaller, the answer gets increasingly accurate, and ultimately it approaches 25%. This agrees with the answer derived from probability theory.


Original Answer:

I’d like to submit 75% as the answer. Here’s my explanation …

Let’s start with a few definitions and assumptions:

- The length of the stick is ‘L’.
- The three pieces are of lengths ‘a’, ‘b’, and ‘c’.
- None of ‘a’, ‘b’, or ‘c’ can be zero (or else there would not be three pieces).
- For the purposes of solving the problem discretely (which I’ll explain later), let’s say the smallest length that any piece can be is ‘d’.
- The triangle must be formed by the three pieces perfectly end to end, with no extra length.
- The probability that you can make a triangle is defined as the number of possible ways to cut up the stick that can make a triangle divided by the number of possible ways to cut up the stick in total. This assumes that each arbitrary way to cut up the stick is equally probable, and that no way is more or less likely than any other.

Now to do the analysis ...

Starting with a stick of length ‘L’, let’s cut off an arbitrary length ‘a’ such that:

0 < a < L

Now, the remaining length of the stick is (L – a). So let’s cut off another arbitrary length ‘b’ such that:

0 < b < (L – a)

Finally, what’s left is ‘c’. Thus:

c = L – a – b

Now let’s try to make a triangle with the three pieces. As several of the previous answerers have noted, this may or may not be possible. In order to make a triangle, the following must be true:

a + b > c

Ok, so what does all this give us? I mentioned earlier that I intend to solve this problem discretely. You’ll also notice in my definitions and assumptions that I imply there is a finite number of ways to cut up the stick. In order to know what this number is, we’ll have to assume a value for ‘d’. In actuality, ‘d’ can be any length at all on a continuum from zero to ‘L’, and thus in fact there is an infinite number of ways to cut up the stick. But in a discrete solution, we assume ‘d’ is some small non-zero value, and by doing so, we can use a computer program to solve the problem by stepping through all the possible combinations in increments of ‘d’. The smaller the value of ‘d’, the more our analysis approximates reality, and the more accurate it will be. This is what is meant by a discrete solution.

Below is a computer program that steps through all the possible combinations and counts the ones that can make a triangle. The probability is the ratio of the number of combinations that can make a triangle to the total number of all combinations. Again, this assumes that each arbitrary way to cut up the stick is equally probable, and that no way is more or less likely than any other. The program is written in old-fashioned Dartmouth BASIC, but it should be pretty straightforward to follow along. Here’s the program:

10 REM This program calculates the probability of being able to make a triangle with 3 arbitrarily cut pieces of a stick.
20 REM Initialize variables:
30 TRIANGLES = 0
40 TOTAL = 0
50 L = 1000
60 D = 1
70 REM Loop through all possible combinations of ‘a’ and ‘b’ in increments of ‘d’:
80 FOR A = D TO (L - (2 * D)) STEP D
90 FOR B = D TO (L – A – D) STEP D
100 C = L – A – B
110 REM Check if this combination can make a triangle
120 REM If so, increment TRIANGLES count:
130 IF A + B > C THEN TRIANGLES = TRIANGLES + 1
140 REM Increment the TOTAL count in all cases:
150 TOTAL = TOTAL + 1
160 REM Loop back to do more combinations of ‘a’ and ‘b’:
170 NEXT B
180 NEXT A
190 REM Print out the results:
200 PRINT “Number of triangles = “; TRIANGLES
210 PRINT “Number of total combinations = “; TOTAL
220 PRINT “Probability of making a triangle =”; 100 * (TRIANGLES / TOTAL)”%”
230 END

In this program listing, I have hard-coded ‘L’ to be 1000 and ‘d’ to be 1. This relates to a stick that is 1000 mm (or 1 meter) long, and therefore ‘d’ is 1 mm. With these values of ‘L’ and ‘d’, I get the following results:

- The total number of possible combinations is 498,501.
- The number of combinations that can make a triangle is 373,751.
- The probability of making a triangle is therefore 373,751 divided by 498,501 … which is about 75%.

The exact number the program gives for the probability is 74.97498%. You can experiment with different values of ‘L’ and ‘d’. Actually, just keeping ‘L’ as 1000 and varying ‘d’ is all that is needed. Generally speaking, as ‘d’ decreases with respect to ‘L’, and therefore more closely approximates the infinite continuum of real physical materials, the more accurate the answer will be. Note that keeping ‘d’ as a number that divides integrally (evenly) into ‘L’ will avoid some error that is due to the discrete nature of the calculation. For example, here are some other results (all with L = 1000):

d = 1 … TOTAL = 498,501 … PROBABILITY = 74.97498%
d = 2 … TOTAL = 124,251 … PROBABILITY = 74.94991%
d = 4 … TOTAL = 30,876 … PROBABILITY = 74.89960%
d = 5 … TOTAL = 19,701 … PROBABILITY = 74.87438%
d = 8 … TOTAL = 7,626 … PROBABILITY = 75.20326%
d = 10 … TOTAL = 4,851 … PROBABILITY = 74.74748%
d = 20 … TOTAL = 1,176 … PROBABILITY = 74.48980%
d = 25 … TOTAL = 741 … PROBABILITY = 74.35898%
d = 40 … TOTAL = 276 … PROBABILITY = 76.08696%
d = 50 … TOTAL = 171 … PROBABILITY = 73.68421%
d = 100 … TOTAL = 36 … PROBABILITY = 72.22222%
d = 125 … TOTAL = 21 … PROBABILITY = 71.42858%
d = 200 … TOTAL = 6 … PROBABILITY = 83.33333%
d = 250 … TOTAL = 3 … PROBABILITY = 66.66667%

Notice that as ‘d’ gets smaller, the probability gets closer to 75% exactly. However, also notice that the answer swings above and below 75% when ‘d’ is larger. This oscillation is an artifact of the discrete calculation in that those values of ‘d’ are not small enough with respect to ‘L’ to give a valid answer.

That all having been said, I’m sure that someone who is proficient in probability theory and/or calculus could come up with a less “brute force” approach. But this is how I approached it. I’d be interested to see if anyone else agrees with my answer or not. In any case, it was an interesting question, and I hope this answer was at least interesting too. Thanks for reading.

Answer 2

Think about it in terms of geometry. If your stick is one meter long, and your cuts are both 1cm from the ends, then the two short pieces will not be long enough to stretch the distance to connect and close the triangle. The minimum length, with two acute angles approaching the limit of 0 degrees and the obtuse angle approaching 180 degrees (minimum area enclosed) would be a cut at 0.25 m and 0.25 m from the ends assuming a 1m total stick length. So the two cuts must equal at least 0.5m when added together, although they can be longer. But as you make them longer you need to keep within the earlier stated bounds. If you're truly arbitrary in where you cut than your probability of falling under this condition would be 1/2. This assumes that you want to lay each piece end to end, with no wasted stick anywhere. If you don't care about waste then the probability truly is 100%.

Addition-
This question was actually bugging me as I was trying to sleep last night, so I did some research this morning. I came up with numerous sites that all show the true answer to be 25%. So, despite what countless SAT and GRE counsellors have told me, my first intuition was wrong. Here's a repost of a good explanation:

In order for the three pieces to form a triangle, all pieces must be shorter than half the length of the original stick. In a triangle, the sum of the lengths of any two sides is always longer than the length of the remaining side. If a piece ends up being longer than half the stick's size, the sum of the other two pieces will be shorter and you will not be able to form a triangle.

To avoid having a piece that is at least half the length of the stick, the two cuts must be on opposite sides of the middle point. The probability of this occurring is ½ (either both cuts end up on the same half or not). Now assume that the first cut is made on point p (where p is a percentage of the full length of the stick). In addition, the second cut must fall below the p + ½ point.The probability for this is of course ½ . Now multiply the two probabilities and you get the final answer: ½ × ½ = ¼

Answer 3

The answer is 0.5 or 50%.

To ensure the triangle can be closed, the two shortest sides added together must be longer than the longest side. After you make the first cut, you are left with a short piece and a long piece. To enable the triangle to be closed, you must take the longest piece for your second cut. This is independent of where the first cut is made.

If you have two pieces, the chance you pick the longer one for the second cut is half, 50%.

(assumes that if the first cut leaves two identical pieces, the resulting 'flat' shape is still considered a triangle.)

Answer 4

You do not necessarily get a triangle if you break a straigh stick in two abritrary places.

However, so long as the sum of the lenghts of the two shortest sticks is GREATER that the longest one, you will ALWAYS be able to make a triangle.

So if you break the sticks into three segmenes, A, B and C, with C being the longest, then in order to get a triangle the following must hold true: A+B > C.

Hope this gives you a good clue.

Aloha.

Answer 5

i made a excel model and i got around 7%. I couldn't help myself i love excel models.

If you want a copy for yourself, send me a message and i can link it to you. No guaranty its right. You need Excel with Analysis Tool pack (this is an add-on); its for the random number generator function. For all i know my model is not the best because it is simplified. It makes one cut off a stick of finite size (1) and stores the left portion as the first length "a", then takes the leftover (right side) and makes another random cut cutting it into two more piece b and c accordingly. It figures out which side is largest and then using IF statements checks to see that the largest piece doesn't exceed the sum of the remaining pieces. If the largest piece size is less then the sum of the remaining pieces a triangle can be made.

Answer 6

You will always get a triangle from 3 pieces of stick.

Answer 7

There were so many woodworking plans with this collection and you will not believe this but there are over thousands plans in the one package deal. Go here https://tr.im/PT6uB
This is really something to find that many all together. For someone like me who is just really starting to get involved with woodworking this was like letting me loose in a candy store and telling me I could have anything I wanted. That was my dream when I was a kid.

Answer 8

Two cuts will give you three pieces so the probability of being able to construct a triangle with them is 100%.

If you are looking for a representation of probabilistic concepts in formal terms, you need to look more closely at the Theory of Probability.

Answer 9

1:1

Answer 10

I would argue that - ALWAYS

2 Cuts give rise to 3 pieces of wood which can be laid down to form TRIANGLES of various sizes!!! :-)