How Many Ways Can You Roll Six Dice with Repeating Numbers?

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    Dice Probability
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Homework Help Overview

The problem involves determining the number of ways to roll six dice such that there are repeating numbers, specifically focusing on scenarios where at least 2, 3, 4, or 5 numbers are the same. The subject area is probability, particularly combinatorial probability related to dice rolls.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of negation to simplify the problem, questioning how to calculate the outcomes for different scenarios of repeated numbers. There are attempts to analyze smaller cases, such as rolling 2 or 3 dice, to build understanding. Some participants express confusion regarding the application of formulas and the organization of possibilities.

Discussion Status

The discussion is ongoing with various approaches being explored. Some participants have suggested breaking down the problem into smaller components, while others are attempting to clarify the reasoning behind certain steps. There is a mix of understanding and confusion, particularly around the application of probability concepts and the organization of outcomes.

Contextual Notes

Participants note the challenge of tabulating possibilities and the complexity of deriving formulas for different cases of repeated numbers. There is an acknowledgment of the need for clearer explanations and specific examples to aid understanding.

  • #61
Shouldn't the probability that all numbers are different plus exaclt 2 are alike and the rest different plus exactly three are alike plus...up to they are all the same be no greater than one?
 
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  • #62
ArcanaNoir said:
Shouldn't the probability that all numbers are different plus exaclt 2 are alike and the rest different plus exactly three are alike plus...up to they are all the same be no greater than one?

Correct, they should sum up to exactly one (or less than one depending on what you calculate exactly).
 
  • #63
Okay, I think I have this problem all done. I've solved it probably more than one way by now, and I got consistent answers. *runs away*
 
  • #64
ArcanaNoir said:

Homework Statement



How many ways can you roll six dice so that at least 2 numbers are the same? At least 3? At least 4? At least 5?

Homework Equations





The Attempt at a Solution



:cry: I've used every equation in the chapter and filled page after page with numbers all moved around. I'm drowning! Please help. I hate probability.

{at least two the same} = E2 + E3 + E4 + E5 + E6, where Ej = {exactly j the same} (and "+" denotes set union). E2 = {2 1s, others all different}+{2 2s, others all different}+... Clearly, all these have the same number of elements, so the number of elements in E2 is 6*|{2 1s, all others different}| (|{.}| = set cardinality). Now think of 6 bins and 6 balls tossed at random into them; we want the number of outcomes in which bin 1 has 2 balls and the others all have 0 or 1 ball each. This the type of thing for which the _multinomial_ distribution was designed.

RGV
 
  • #65
ArcanaNoir said:
*runs away*

Hey! Come back! :smile:
There's more...
 

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