Probability of Rolling Dodecahedra Outcomes

Overview of Dodecahedra and Probability Calculation

  • Dodecahedra are polyhedra with 12 faces, used in various games and probability scenarios.
  • In this probability problem, we are rolling 15 dodecahedra and need to find the specific probability of a particular distribution of outcomes.

Problem Statement

  • Total Dodecahedra Rolled: 15

  • Desired Outcomes:

    • Exactly one (1) roll of 9
    • Exactly two (2) rolls of 8
    • Exactly two (2) rolls of 7
    • Exactly two (2) rolls of 4

  • This represents a total of 1 + 2 + 2 + 2 = 7 faces accounted for, with the remaining 15 - 7 = 8 dodecahedra yielding any number from 1 to 12 except 9, 8, 7, and 4.

Calculation of Probability

Step 1: Total Outcomes

  • Each dodecahedron has 12 possible outcomes (faces numbered from 1 to 12).

  • Therefore, the total number of outcomes when rolling 15 dodecahedra is given by:

    extTotalOutcomes=1215ext{Total Outcomes} = 12^{15}

Step 2: Consideration of Specific Outcomes

  • To compute the probability of obtaining exactly the desired distribution, we need to consider the arrangement of successes (the desired outcomes).
  • The arrangement of outcomes can be calculated using multinomial coefficients.
Multinomial Coefficient Calculation

  • The formula for the multinomial coefficient for grouping outcomes is:

    n!k<em>1!imesk</em>2!imesk<em>3!imesk</em>4!\frac{n!}{k<em>1! imes k</em>2! imes k<em>3! imes k</em>4!}

  • In our case, let:

    • n = 15 (total dodecahedra)
    • k_1 = 1 (for the one 9)
    • k_2 = 2 (for the two 8's)
    • k_3 = 2 (for the two 7's)
    • k_4 = 2 (for the two 4's)
    • Remaining k_5 = 15 - (1 + 2 + 2 + 2) = 8 (for other numbers)

  • Therefore, the multinomial coefficient is:

    15!1!imes2!imes2!imes2!imes8!\frac{15!}{1! imes 2! imes 2! imes 2! imes 8!}

Step 3: Counting the Outcomes of Non-Specific Rolls

  • The remaining 8 dodecahedra can roll any number except 9, 8, 7, and 4. This provides us with the numbers: 1, 2, 3, 5, 6, 10, 11, and 12 (a total of 8 outcomes).

  • The number of possible outcomes for the remaining rolls is therefore:

    888^8

Step 4: Final Probability Calculation

  • The probability of getting exactly the desired combinations can now be calculated by combining all components:

    P=15!1!imes2!imes2!imes2!imes8!imes881215P = \frac{\frac{15!}{1! imes 2! imes 2! imes 2! imes 8!} imes 8^8}{12^{15}}

  • This expression encapsulates the ratio of the favorable outcomes to total outcomes of rolling the dodecahedra.

Conclusion

  • The final expression gives the probability of achieving the desired distribution when rolling 15 dodecahedra, accounting for specific outcomes and the arrangements required for such outcomes.