Aaron Low

How: Balancing the deck

 

1533939 = Sample space of combinations of 5 card hands (47C5)

This will be used a lot....

 

As with any card game, the numbers matter. Choosing those numbers at times can feel arbitrary at best. But with some basic philosophy and discrete math, we can easily determine the outcomes. 

 

Police card x 18:

The police cards are how the game ends. So there had to be a considerable amount of cards in the deck. If we added too many cards, we would end up making the win state impossible. So I used some combination math and basic math to determine some goals. 

 

I decided the sweet spot would be around 18, for a 38% chance to draw. 

 

To calculate the odds of 3 police cards in a hand: 

 

816 = Combination of 3 police cards in a pool of 18 (18C3)

406 = Combination of 2 other cards of 29 (29C2)

331296 = Combinations of 3 police and 2 other cards

 

331296/1533939 = .2159

 

This gives us a decent prediction of how often a player will have 3 police cards in their hand. That being 21.59%, which I believe to be a decent rate to force an action for a player. 

 

Permutations do not matter because this game doesn't consider the order. 

 

Safe Peek Cards x 8:

Basically, a get out of jail free card. So these cards needed to be limited. Giving players too many, they could safely peek every time. So we found the sweet spot to be 8 cards. With a draw rate of 17%, it forced players to make harder decisions when it comes to seeing in the deck. 

 

28 = Combinations of 2 Safe peek cards of 8

9139 = Combinations of 3 other cards  

255892 = Combinations of 2 Safe Peek and 3 other cards
255892/1533939 = .1668

 

So a 16% chance of a player having TWO safe peeks. So not too often.
 

Part Cards  x 11:

The win state of the cards. I won't keep spewing math at you, its made its point. Cards have a descending repetition of useful cards in the deck. 

 

Part 3 has 6 cards in the deck = 12% Draw Chance

Part 2 has 3 cards in the deck =  6% Draw Chance

Part 1 has 2 cards in the deck =  4% Draw Chance

 

While is attractive to assume we can combine those percentages to get a chance. It is highly likely each player will have one Part 3 card in a drop box by the time it actually matters.  

 

Trade Cards x 5:

Trade cards are one of the most powerful cards in stopping a win state. You can trade out a card and steal from their dropbox. Once they are used they cannot be redrawn into a hand again.

 

Safe Draw Two Cards x 5:

Safe Draw allows the player to increase their potential hand which gives them more options and combinations to create a win state. It specifically states you cannot draw a police card as an added benefit of it.