Tips piratenkapern

For the rules of piraten kapern, see GameHelpPiratenKapern

Probabilities

Binomial formula

   n!    × pᵏ(1 − p)ⁿ⁻ᵏ
k!(n-k)!

n: number of trails (dice thrown)
k: number of successes (dice with a face value)

p: probability of success (of a die face value)

Example

Probability of throwing 3​ with 5 dice:

   5!    × (⅙)³ × (1 − ⅙)⁵⁻³
3!(5-3)!

=  5×4×3×2×1  × (⅙)³ × (⅚)²
  3×2×1 × 2×1

= 10 × (⅙)³ × (⅚)²

≈ 0.0321 or 3.21%

2 dice

Probabilities of throwing X skulls with two six-sided dice

In words	In maths	Percentage
Probability of no skulls	P(X = 0) = (⅚)²	≈ 69.4%
Probability of one skull	P(X = 1) = 2 × (⅙) × (⅚)	≈ 27.8%
Probability of two skulls	P(X = 2) = (⅙)²	≈ 2.78%

8 dice

Probabilities of throwing X skulls with eight six-sided dice

In words	In maths	Percentage
Probability of no skulls	P(X = 0) = (⅚)⁸	≈ 23.3%
Probability of one skull	P(X = 1) = 8 × (⅙) × (⅚)⁷	≈ 37.2%
Probability of two skulls	P(X = 2) = 28 × (⅙)² × (⅚)⁶	≈ 26.0%
Probability of three skulls	P(X = 3) = 56 × (⅙)³ × (⅚)⁵	≈ 10.4%
Probability of four skulls	P(X = 4) = 70 × (⅙)⁴ × (⅚)⁴	≈ 2.60%

Probabilities of throwing X or more skulls with eight six-sided dice

In words	In maths	Percentage
Probability of one or more skulls	P(X ≥ 1) = 1 − P(X = 0) = 1 − (⅚)⁸	≈ 76.7%
Probability of two or more skulls	P(X ≥ 2) = 1 − [ P(X = 0) + P(X = 1) ] = 1 − [ (⅚)⁸ + 8 × (⅙) × (⅚)⁷ ]	≈ 39.5%
Probability of three or more skulls	P(X ≥ 3) = 1 − [ P(X = 0) + P(X = 1) + P(X = 2) ] = 1 − [ (⅚)⁸ + 8 × (⅙) × (⅚)⁷ + 28 × (⅙)² × (⅚)⁶ ]	≈ 13.5%
Probability of four or more skulls	P(X ≥ 4) = 1 − [ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) ] = 1 − [ (⅚)⁸ + 8 × (⅙) × (⅚)⁷ + 28 × (⅙)² × (⅚)⁶ + 56 × (⅙)³ × (⅚)⁵ ]	≈ 3.07%