Roll 6 Dice: Solve Probability Puzzle!

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Discussion Overview

The discussion revolves around the probability of rolling six dice and obtaining all six different numbers in a single roll. Participants explore the mathematical reasoning behind the expected number of rolls required to achieve this outcome.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant asks how many times one needs to roll six dice to see all six different numbers.
  • Another participant suggests an average of 64.8 rolls, although they humorously note the imprecision of the decimal.
  • A third participant agrees with the 64.8 estimate and provides a mathematical justification involving permutations and probabilities, referencing the Poisson distribution to explain the average interval between successful rolls.

Areas of Agreement / Disagreement

There is a general agreement on the estimate of 64.8 rolls, but the discussion includes some light-hearted disagreement regarding the precision of the decimal.

Contextual Notes

The mathematical reasoning presented relies on specific assumptions about the distribution of outcomes and the calculations of probabilities, which may not be universally accepted without further verification.

Raybert
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On average, how many times do you need to roll six dice together to see all six different numbers turn up within a single such group roll?
 
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64.8 times?
 
CompuChip said:
64.8 times?

I agree, except for the .8 :)
 
I agreed with 64.8, for the following reasons:

There are 6! ways of throwing all six values and 66 possible results, so the probability in each throw is 6!/66 = (1*2*3*4*5*6)/(6*6*6*6*6*6) = (4*5)/(6*6*6*6) = 5/(3*3*6*6) = 5/324.

The average interval between such throws (or, as in this case, before the first such throw) is therefore the reciprocal of this, 324/5 = 64.8, as usual for a Poisson distribution.
 

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