p&c q16

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Saumyojit

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If n identical dice are rolled, then number of possible outcomes are​



I don't want the solution right now .
What is the meaning of identical dice. Every dice will be same looking ? What is the significance ?


Outcomes are 1,2 ; 1,1 ; 1,3 ; ..... and another set might be 2,1 ; 2,2 ; 2,3 ....

1,2 is the same as 2,1 or different?
 
Saumyojit said:
If n identical dice are rolled, then number of possible outcomes are
I don't want the solution right now .
What is the meaning of identical dice. Every dice will be same looking ? What is the significance ?
Outcomes are 1,2 ; 1,1 ; 1,3 ; ..... and another set might be 2,1 ; 2,2 ; 2,3 ....
1,2 is the same as 2,1 or different?
Click to expand...
You clearly need to read to read upon dice in this link.
Please read it and then post
 
Saumyojit said:

If n identical dice are rolled, then number of possible outcomes are​



I don't want the solution right now .
What is the meaning of identical dice. Every dice will be same looking ? What is the significance ?


Outcomes are 1,2 ; 1,1 ; 1,3 ; ..... and another set might be 2,1 ; 2,2 ; 2,3 ....

1,2 is the same as 2,1 or different?
Click to expand...
I'm not sure what they mean; my first impression is that they may be using "identical" to mean "indistinguishable", in which case order would not matter (and outcomes would not be equally likely). But I'm not sure; ideally I'd be looking for indications in the context, if I had the source. One clue I'd look for is whether the author might be trying to introduce this idea that outcomes do not necessarily have to be equiprobable.

In well-written math problems, either standard terms like "indistinguishable" should be used, or they should explain as fully as possible in everyday terms (e.g. adding "if we can't tell which die has which number, ...").

Incidentally, whether the dice are distinguishable does not affect probability, only what outcomes can be distinguished. So for most problems that start this way, there would be no need to think about what "identical" means!
 
Saumyojit said:
1,2 is the same outcome as 2,1 or different?
Click to expand...
If you toss two ordinary standard dice the outcomes are the sum of the numbers showing on each die.
Those outcomes are [imath]2=1+1,~3=1+2+1,~4=1+3=3+1=2+2,~\cdots~12=6+6[/imath]. Those are eleven outcomes.
In terms of thirty six ordered pairs [imath]\{(m,n): 1\le m\le 6, 1\le n\le 6\}[/imath].
Look at this expansion:
[imath]\left(x+x^2+x^3+x^4+x^5+x^6\right)^2=x^{12} + 2 x^{11} + 3 x^{10} + 4 x^9 + 5 x^8 + 6 x^7 + 5 x^6 + 4 x^5 + 3 x^4 + 2 x^3 + x^2[/imath]
The term [imath]5x^8[/imath] tells us that tossing two dice, the sum 8 appears in 5 ways so its probability is [imath]\dfrac{5}{36}.[/imath]
If we toss four standard dice the probability of getting an eighteen sum is [imath]\dfrac{80}{6^4}[/imath]. WHY?
 
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pka said:
If you toss two ordinary standard dice the outcomes are the sum of the numbers showing on each die.
Click to expand...
This raises another question: What do they mean by "outcome" in the first place? I would not (without context) take it to mean the sum, in part because that is too easy, and in part because there are many problems that start the same way, in which the sum is not in view: They could end up asking for the probability that the difference is 4 (assuming n = 2), or (more generally) that all numbers are even, or that their product is a multiple of 3, or that they can form an arithmetic progression.

On the other hand, I don't know why one would ask for the number of outcomes where an outcome is an unordered pair. We generally define the outcomes of an experiment in whatever way works best for a particular problem, and I can't think of any where that would be appropriate.

But I can't say for sure what it does mean.

Once again, I would want to know who originally wrote the problem, in order to see the context. And I strongly suspect this comes from one of those sites where people submit problems with no context, and people answer based on whatever they think is right. I would ignore such sites.

The problem is either poorly stated, or quite difficult, or both. I would not pursue it.
 
Dr.Peterson said:
The problem is either poorly stated, or quite difficult, or both. I would not pursue it.
Click to expand...
math.stackexchange.com

How many are the possible outcomes from throwing $n$ (identical) dice

How many are the possible outcomes from throwing $n$ (identical) dice. I know it is a combination with repetition, but don't really know how to apply the formula. I need an explanatory answer wit...
math.stackexchange.com math.stackexchange.com
please don't see the answer .



Dr.Peterson said:
This raises another question: What do they mean by "outcome" in the first place? I would not (without context) take it to mean the sum, in part because that is too easy,
Click to expand...
Why outcome means sum of individual outcome of each dice.

pka said:
Look at this expansion:
(x+x2+x3+x4+x5+x6)2=x12+2x11+3x10+4x9+5x8+6x7+5x6+4x5+3x4+2x3+x2\left(x+x^2+x^3+x^4+x^5+x^6\right)^2=x^{12} + 2 x^{11} + 3 x^{10} + 4 x^9 + 5 x^8 + 6 x^7 + 5 x^6 + 4 x^5 + 3 x^4 + 2 x^3 + x^2(x+x2+x3+x4+x5+x6)2=x12+2x11+3x10+4x9+5x8+6x7+5x6+4x5+3x4+2x3+x2
The term 5x85x^85x8 tells us that tossing two dice, the sum 8 appears in 5 ways
Click to expand...
okay .
if we assume outcome means sum of individual outcome of each dice.

then, rightly so , the sum 8 appear in 5 ways as coefficient of x^8 is 5 . (Multinomial theorem )
 
Saumyojit said:
Why outcome means sum of individual outcome of each dice.
Click to expand...
I said I wouldn't interpret it that way here. It could be taken that way in the context of a problem about the probability of a sum, but in other contexts it would not. Why do you think I said it does mean this??

Saumyojit said:
please don't see the answer .
Click to expand...

Having replied to you, I now look at the page you refer to (which is not one of the pages I found before, since it is not exactly the same question you asked). Without scrolling down to see the answers, I see first that this (different) question does come from a book (not identified, and with no context provided), and that the answer is not what you would say if you interpret the question as being about sums. I also see that the immediate response was to ask what the question means, and point out that "identical" could mean different things. So far, they are in agreement with me.

Then I see that the one actual answer accepts the assumption that the question is about combinations with repetition, which is true if we take it the way I initially said to take it. Since you said not to read that, I will not comment further; if you want, you can ask why the formula shown applies to combinations with repetition.
 
Dr.Peterson said:
combinations with repetition
Click to expand...
So, identical means indistinguishable

Now what is the approach ?

I know partion mathod using bars .But how to use it here ?

Suppose there are three dices and five dividers are taken as each dice contains six values . So i am dividng the dice into six distinct values.

SO, ||D| D D || = suppose This arrangement tells us that first dice contains 3 as a value and the other two dices has 4 as a value and that is how i am counting the possible outcomes of n dices .

1,2,1= 2,1,1= 1,1,2 . Identical

n+5 C 5
 
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Saumyojit said:
So, identical means indistinguishable
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That's not actually what I said. I said this is one likely interpretation of "identical", but not a technical meaning, so I would prefer clarification.

Saumyojit said:
I know partion mathod using bars .But how to use it here ?
Click to expand...
I think after you asked this question, you answered it for yourself, so you don't need any more help?

Your explanation is nice, actually clearer than how I initially worked it out.
 
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