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# 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%