# Gang of Four

Adapted from the card games of ancient China, Gang of Four^{™} has emerged as one of the most popular online card games to come out of Asia in decades. Simple to learn and fun to play, Gang of Four is filled with intriguing strategies. To win, be the first to rid yourself of all your cards, using similar combinations as those found in Poker. But beware – a single misstep may find you in a struggle to survive!

“It is rare that a game presents players with this complete, challenging and enjoyable experience.”

The Games Journal

## Probabilities & shuffle

What is the probability of having a Gang of Four combination after a well randomized shuffle? This probability is rather high: almost 30%! Here is why.

The rule to apply is the "hypergeometric law", which is a heavily used in surveys. We are dealing with a random drawing without putting items back.

Let's consider a bag that contains N balls where a number 'a' of them are red. We draw 'n' balls. The probability to get 'k' red balls is:

with:

(n! meaning n x (n-1) x (n-2)... Example: 7!=7x6x5x4x3x2x1)

Applied to Gang of Four, this gives for a Gang of twos (N=64, a=6, k=4, n=16):

If you add up all probabilities (which is a good approximation, because the events are almost independent), you get:

Another interesting element is to compute the probability that the other players have one or more Gang of Four combinations when you have one in your hand

Number of Gang of Four for other players | ||||||

0 | 1 | 2 | 3 | 4 | ||

knowing that your hand contains: | 0 | 35.8% | 39.7% | 18.7% | 5.0% | 0.7% |

1 | 31.3% | 39.2% | 21.8% | 6.4% | 1.1% | |

2 | 26.3% | 37.5% | 25.2% | 8.8% | 1.8% | |

3 | 9.1% | 31.8% | 31.8% | 22.7% | 4.6% |