Looking at how to make use of binomial coefficients

 

Good for tabletop games

 

Binomial Distribution

 

Binomial distribution - probability curve using two curves

 

Formula that allows us to answer the following type of design questions

• How many X are likely to Y? (How many sushi rolls will be eaten from the batch today)

• Are these numbers balanced and/or correct?

 

Probability of getting exactly X successes from N attempts =

The number of ways we can choose X (from N) (N over X or the binomial coefficient)

AND (multiplied by)

The probability of getting X (‘probability’) that number of times (X times) (p to the power of x)

AND (multiplied by)

The probability of not getting X (1-p) the rest of those times (N-X) (1-p to the power of N-X)

 

(P(X)) = (N over X)p to the power of x(1-p)tothepowerof(N-X)

 

Example: Merfolk vs Pirates

 

The game (combat)

D6 roll over system

Each character has:

• 1 wound

• To hit roll

• To wound roll

• Save roll

If the attacking character succeeds at their to hit roll AND succeeds at their to wound roll AND the defending character fails their save roll, then the defending character takes 1 wound. Anything else fails.

 

The pirates

To hit (H) : 5+

To would (D) : 3+

To save (S) : 6+

 

The merfolk

To hit (H) : 3+

To wound (D) : 5+

To save (S) : 4+

 

Probability that a pirate could kill a merfolk = Pirate must hit AND cause damage AND the merfolk fails their save

=

5+ AND 3+ AND <4

=

2/6 AND 4/6 AND 3/6

=

24/216 = 11% or 1/10

 

Probability that a merfolk could kill a pirate

=

3+AND 5+ AND 6+

=

4/6 AND 2/6 AND 5/6

=

40/216 = 19% = 1/5

 

Game unbalanced. Way to balance would be to double the number of pirates to merfolk

 

 

Own game scenario - Angels vs demons

 

Imps are fighting angels at the gates of hell. At any given time there is a 10% chance that a team is doused in hellfire, reducing their attack and defense by 50%. Angels will have a 10% chance of an insta kill upon hitting

 

Combat system: Roll over D12

1 hit point (cant fly with one wing)

Roll to hit

Roll to do damage

Roll to dodge

 

Angels

To hit (H) : 9+ - Careful shooters, high accuracy

To damage (D) : 5+ - Not as ruthless as the demons, low chance of damage

To dodge (S): 7+ - Large wings and little bodies = Slow

 

Demons

To hit (H) : 5+ - Wild shooters, low accuracy

To damage (D) 9+ - Stronger than angels, high chance of damage

To dodge (S) 3+ - Can fly faster and are more flexible

 

No modifiers:

 

Angel hitting a demon = Angel must hit x do damage x demon must fail save

= 5/12 x 9/12 x 3/12

= 5/64 = 0.078125 = 7% or 1/7

 

 

Demon hitting an angel = 5/12 x 9/12 x 3/12

= 9/12 x 5/12 x 7/12

= 35/195 = 0.1822917 = 18% or 1/5

 

 

To balance chances there should be 2.5x angels:demons

 

 

With modifiers

 

Angels chance of hitting and insta death = 10% of 7% = 0.7% chance

 

Chances with hellfire:

 

Angels = 3.5% chance of hitting, 36% chance of being hit

Demons = 9% Chance of hitting, 14% chance of being hit

 

 

Binomial distribution (without modifiers)

For the sake of example there are 5 demons and 15 angels

P(X) = (N¦X) p^X 〖(1-p)〗^((N-X))

 

Likelihood of hitting a demon

 

N = 15

P = 1/7

1-P = 6/7