Dice Numbers Creating Triangles

Probability Level 5

Potsawee has a fair six-sided die. He throws the die 3 times, and the numbers shown on the upper side each time are $a,b,c$ respectively. Find the probability that the lengths $(a,b,c)$ are able to create an isosceles triangle . Give your answer to three significant figures.

The answer is 0.319.

2 solutions

Jessica Wang
Jul 19, 2015

First of all, there are $6\times 6\times 6=216$ total values of $a,b,c$ .

When the lengths $a,b,c$ are able to create an isosceles triangle, we can divide it into two different situations:

$[1]. \; a=b=c$ : We have $6$ values.

$[2].$ There are exactly two sides which are of the same length. Without loss of generality, assume $a=b\neq c$ .

When $c=1$ , $a=b=2,3,4,5,6$ ;

When $c=2$ , $a=b=3,4,5,6$ ;

When $c=3$ , $a=b=2,4,5,6$ ;

When $c=4$ , $a=b=3,5,6$ ;

When $c=5$ , $a=b=3,4,6$ ;

When $c=6$ , $a=b=4,5$ .

We have $21$ values of $a,b,c$ .

Therefore, the probability required is $\frac{6+3\times 21}{216}\approx \boxed{0.319}$ .

Nice and simple solution; I'm really bothered I forgot the equilateral triangles.

Tiago Abi-Ramia - 5 years, 10 months ago

The question doesn't actually say to include them, and not all definitions of isosceles include equilateral triangles. Therefore, you shouldn't be bothered.

Axelrod Polaris - 5 years, 10 months ago

@Axelrod Polaris Thank you for the reminder. I have added a link to the "isosceles triangle" in the question, which gives a definition.

Jessica Wang - 5 years, 10 months ago

Should not the number of ways of obtaining 2 equal faces with 3 dices be 90?

guido barta - 5 years, 4 months ago

Bill Bell
Aug 3, 2015

from itertools import product

def isIsosceles(a,b,c):
    a,b,c=sorted([a,b,c])
    if a==b: 
        return a+b>c
    elif b==c: 
        return True
    else:
        return False

overallCount=0
specificCount=0
for a,b,c in product(range(1,7),range(1,7),range(1,7)):
    overallCount+=1
    specificCount+=isIsosceles(a,b,c)
print float(specificCount)/overallCount

Dice Numbers Creating Triangles

The answer is 0.319.

2 solutions

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