Can You Simplify Probability Calculations for Multiple Dice Rolls?

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SUMMARY

The discussion centers on calculating the probability of the sum of three rolled dice being less than or equal to 9. Participants suggest using combinatorial methods instead of enumerating all 216 possible outcomes. A specific approach involves calculating the probabilities of two dice and then incorporating the third die's outcomes. The example provided illustrates the calculation for rolling a total of 3 using the formula P(first two=2) x P(last die=1) = (1/36)(1/6) = 1/216, demonstrating a systematic method for probability determination.

PREREQUISITES
  • Understanding of basic probability concepts
  • Familiarity with combinatorial mathematics
  • Knowledge of rolling dice probabilities
  • Ability to perform calculations involving fractions
NEXT STEPS
  • Research combinatorial probability techniques for multiple events
  • Learn about the probability distribution of rolling multiple dice
  • Explore the concept of generating functions in probability
  • Study advanced probability topics such as conditional probability and independence
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Students studying probability, educators teaching combinatorial mathematics, and anyone interested in simplifying complex probability calculations involving dice.

ArcanaNoir
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Homework Statement



"Find the probability of the sum of three rolled die being less than or equal to 9."

Given problems like this, my question is: is there some way to find the answer besides writing out all 216 combinations and then counting the ones that equal 9? I feel like there is, but I'm just being blind.
 
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You could consider the total as being the sum of two dice and the third. Presumably, you know the probabilities involved with rolling two dice.

For example, the probability of rolling a 3 would be P(first two=2)xP(last die=1) = (1/36)(1/6) = 1/216.
 
Ah, that looks good. Thanks!
 

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