Probability of getting arithmetic sequence from 3 octahedron dice

[songoku]

Homework Statement:

Please see below

Relevant Equations:

Probability

Arithmetic Sequence

I try to list all the possible sequences:
1 2 3
1 3 5
1 4 7
2 3 4
2 4 6
2 5 8
3 4 5
3 5 7
4 5 6
4 6 8
5 6 7
6 7 8

I get 12 possible outcomes, so the probability is $\frac{12 \times 3!}{8^3}=\frac{9}{64}$

But the answer key is $\frac{5}{32}$ . Where is my mistake? Thanks

 

[haruspex]

Homework Statement:: Please see below
Relevant Equations:: Probability

Arithmetic Sequence

Where is my mistake?

In the sequence a, a+b, a+2b, what are all the possible values of b?

 

[songoku]

In the sequence a, a+b, a+2b, what are all the possible values of b?

I think for this case the difference should be positive integer so b can be 1, 2, or 3

 

[haruspex]

I think for this case the difference should be positive integer so b can be 1, 2, or 3

I disagree. 1, 1, 1 is a perfectly good arithmetic sequence.

 

[Delta2]

I know I ll spoil abit the solution but in order to be a bit more formal and since the dice is 8-ply it will have to be $$1\leq a+2b\leq8\Rightarrow \frac{1-a}{2}\leq b\leq \frac{8-a}{2}$$ and ofc $$b\geq 0$$.

 

[Delta2]

Anyway I think you did find the b correctly, except you didn't took the case b=0 (and tbh I myself didn't think of that). If you add the 8 cases (a,a,a) ,a=1...8 to your result, you get the answer key.

 

[songoku]

I disagree. 1, 1, 1 is a perfectly good arithmetic sequence.

I know I ll spoil abit the solution but in order to be a bit more formal and since the dice is 8-ply it will have to be $$1\leq a+2b\leq8\Rightarrow \frac{1-a}{2}\leq b\leq \frac{8-a}{2}$$ and ofc $$b\geq 0$$.

Is 1, 1, 1 can also be called geometric sequence?

 

[Delta2]

Is 1, 1, 1 can also be called geometric sequence?

Yes it is arithmetic sequence with $\omega=0$ and geometric sequence with $\omega=1$.

 

[songoku]

Yes it is arithmetic sequence with $\omega=0$ and geometric sequence with $\omega=1$.

How about 0, 0, 0? Can that also be called both arithmetic and geometric sequence?

 

[Delta2]

I think yes but why are you asking these questions...

 

[Delta2]

0 isn't a possible number from the dice.

 

[songoku]

I think yes but why are you asking these questions...

I just want to know so if I do other questions I know which one I can consider as arithmetic or geometric sequence.

Thank you very much for the help and explanation haruspex and Delta2