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# Triangle probability

## Question

If the side of a triangle is dicided by throwing a dice 3 times. Then finding probability of forming Isoceles Triangle, is

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Excluding equilateral triangles

2/9

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3 in 20

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This sounds like it might be a homework problem, so I won’t flat out answer it but will give a hint on how to approach it.

Spoiler

It becomes a lot easier if you solve for the probability that you roll the dice three times and NOT get an isosceles triangle. In other words: what's the probability that you roll three times, and on each throw you roll a number that hasn't appeared yet?

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In a nutshell ...

Spoiler

There are 216 possible outcomes with three dice forming ...

Equilateral triangles: 6/216 (1/36)

No isosceles triangles: 120/216 (5/9)

Acute isosceles triangles: 45/216 (5/24)

Obtuse isosceles triangles: 45/216 (5/24)

All isosceles triangles (including equilateral triangles): 96/216 (4/9)

Edited by rocdocmac
typo
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A couple of questions:

Is a side of the triangle determined by a single die roll?  Or do you sum up three rolls per side?

Are non-possible triangles considered valid solutions?  e.g. A roll of 1,1,6 might appear isosceles, but would not be a valid triangle.

Edited by BobbyGo
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On 5/8/2019 at 10:57 PM, Thalia said:

Excluding equilateral triangles

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2/9

7/24 without equilateral

3/8 with equilateral

Both excluding impossible triangles.

Edited by Thalia
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This gets tricky with rolls like (1, 1, 6) that look isosceles but don't actually form triangles.

How would you handle such rolls? Would you omit rolls that don't form a triangle from analysis (equivalent to saying that if you get a roll that doesn't form a valid triangle then roll again), or count them as a failure to form an isosceles triangle? And would you consider a "straight line" throw like (3, 3, 6) to be a valid triangle albeit with zero area or nah?

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Revised ...

Spoiler

I was way out with my first attempt!

Equilateral (EL) = 2/72 = (1/36)

Acute isosceles (AI) =  16/72 (2/9)

Obtuse isosceles (OI) =  5/72

Impossible triangles = 9/72 (1/8)

Normal triangles = 40/72 (5/9)

Total isosceles (AI+OI) = 18/72 (1/4)

Including equilateral (AI+OI+EL) = 23/72

Edited by rocdocmac
typo
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Spoiler

Here's a visual representation including and excluding "impossible" triangles

You can make your own subtotals!

Edited by rocdocmac
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On 5/11/2019 at 4:26 PM, Thalia said:

Hide contents

7/24 without equilateral

3/8 with equilateral

Both excluding impossible triangles.

Well, I seem to have done some weird math when adding equilateral. New result including equilateral:

23/72

I think rocdocmac and I agree on the figure including equilateral now but we seem to have diverged on the other figure...

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My revised answers sent in previously were not all correct. These should be fine ...

Spoiler

With impossible triangles/straight-line cases:

Equilateral (EL) = 2/72 (1/36)

Acute isosceles (AI) =  15/72 (5/24)

Obtuse isosceles (OI) =  6/72 (1/12)

Impossible triangles (including "straight line" cases) = 9/72 (1/8)

Normal non-isosceles triangles = 40/72 (5/9)

Total isosceles (AI+OI) = 21/72 (7/24)

Including equilateral (AI+OI+EL) = 23/72﻿

Without impossible triangles/straight-line cases:

Equilateral (EL) = 2/63

Acute isosceles (AI) =  15/63 (5/21)

Obtuse isosceles (OI) =  6/63 (2/21)

Normal non-isosceles triangles = 40/63

Total isosceles (AI+OI) = 21/63 (1/3)

Including equilateral (AI+OI+EL) = 23/63﻿

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Excluding straight lines, impossible triangles and non-isosceles triangles, the distribution of all isosceles triangles is as follows ...

Spoiler

Edited by rocdocmac
typo

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