How Do You Calculate the Probability of a Sum in n Dice Rolls?

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Discussion Overview

The discussion centers around calculating the probability of obtaining a specific sum, denoted as r, from the rolls of n dice. Participants explore the possibility of formulating a probability distribution to assess various conditions related to the sum, such as greater than r, equal to r, and less than r. The conversation includes considerations of combinatorial methods and generating functions as potential approaches to generalize the problem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about calculating the probability of a sum of r in n dice rolls and expresses a desire for a general method rather than specific cases.
  • Another participant suggests that the probability can be calculated as the number of successful outcomes divided by the total possible outcomes, affirming that a probability distribution can be established.
  • A participant mentions that while specific cases can lead to generalizations, the process may not be straightforward and suggests revisiting combinatorial concepts.
  • One participant proposes a method involving generating functions, specifically referencing the product of terms representing the outcomes of dice rolls, and notes that the coefficient of a term corresponds to the number of ways to achieve a specific sum.
  • There is a suggestion to consult additional resources for combinatorial concepts, indicating that foundational knowledge in combinations and permutations is necessary.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and approaches to the problem, with no consensus reached on a singular method or solution. Multiple perspectives on how to tackle the calculation of probabilities and the use of combinatorial techniques remain present.

Contextual Notes

Participants highlight the need for a generalizable approach and the potential complexity of the problem, indicating that foundational knowledge in combinatorics may be essential for further exploration.

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How does one calculate the probability of a sum of r in the dice rolls of n dice? Can a probability distribution be written for something like this, to calculate the probability of a sum greater than r, greater than or equal to r, equal to r, less than or equal to r, less than r, etc.?

I do not want to simply brainstorm solutions leading to a sum of r, unless it is easy to generalize from there to n dice (which, from my consideration of n=2 and n=3, doesn't seem like the case - I need a more general way of thinking about it).
 
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How does one calculate the probability of a sum of r in the dice rolls of n dice?
Same way you calculate any probability - the number of ways it can happen divided by the total number of things that can happen.

Can a probability distribution be written for something like this, to calculate the probability of a sum greater than r, greater than or equal to r, equal to r, less than or equal to r, less than r, etc.?
Yes.

I do not want to simply brainstorm solutions leading to a sum of r, unless it is easy to generalize from there to n dice (which, from my consideration of n=2 and n=3, doesn't seem like the case - I need a more general way of thinking about it).
The more general way is above, and going through specifics does generalize eventually - it's just not so obvious. Perhaps revisit combinatorics?
 
Simon Bridge said:
Same way you calculate any probability - the number of ways it can happen divided by the total number of things that can happen.

Yes.

The more general way is above, and going through specifics does generalize eventually - it's just not so obvious. Perhaps revisit combinatorics?

What should I use to revisit combinatorics properly? I've read the Schaum's Outline but it doesn't seem to cover problems of this standard.
 
Schaum probably has enough information to get you going - you only need the basic concepts: the bit before "combination" and "permutation" notation is defined.
OR you can just http://www.mathpages.com/home/kmath093.htm and see how others have done it...
 
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There is a general method related to the combinatorics of the problem. Consider the product:

(x + x^2 + x^3 + x^4 + x^5 + x^6)^3

If you multiply it out and combine like terms then, for example, the coefficient of x^4 gives the number of sequences of 3 die rolls that produces a sum of 4.

This method is an example of using "generating functions" in combinatorics The term "generating function" has different meanings in different branches of math, so to search for the topic on the web, you should use a more specific search that the keywords "generating function". There are many examples on the web of using this type of generating function as a step in solving problems like "How many ways can you pay a dollar debt if you have 3 quarters, 5 dimes, and 6 nickels".
 

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