Yahtzee

Probability Puzzles in R

Peter Chi

Assistant Professor of Statistics Villanova University

Yahtzee scoring

Total number of possibilities:

$$ n_1 \times n_2 \times \ldots \times n_k $$

Example. Roll three dice. Total number of configurations:

$$ 6 \times 6 \times 6 = 6^3 $$

6^3

$k$ objects, $n$ total possibilities, Each possibility used once at most

Total number of configurations:

$$ n \times (n-1) \times ... \times (n-k+1) = \frac{n!}{(n-k)!} $$

Example. Number of ways for three dice to land as {2,3,4}:

$$ 3 \times 2 \times 1 = \frac{3!}{(3-3)!} = 3! $$

factorial(3)

Given disjoint events $A$ and $B$::

$$ P(A \cup B) = P(A) + P(B) $$

Example 1. Probability of rolling {2,3,4} or {3,4,5} with three dice

factorial(3)/6^3 + factorial(3)/6^3

0.05555556

Example 2. Probability of rolling three dice landing on the same denomination

1/6^3 + 1/6^3 + 1/6^3 + 1/6^3 + 1/6^3 + 1/6^3

0.02777778

$n$ total objects Choose $k$ of them; order does not matter

Total number of ways:

$$ {n \choose k} = \frac{n!}{k! \times (n-k)!} $$

Example. Number of ways to choose 2 dice out of 3:

$$ {3 \choose 2} = \frac{3!}{2! \times (3-2)!} = 3$$

choose(3,2)

Example. Roll 10 dice

Number of ways to roll 5 of one denomination and 5 of another:

n_denom <- factorial(6) / factorial(4)

n_groupings <- choose(10,5) * choose(5,5)

n_total <- n_denom * n_groupings
n_total

Probability Puzzles in R