I am developing a 4 players AI based hearts game. Every player has random 13 cards of the 52 cards. I need some probabilities to make the AI.
- In the first move (Everyone has 13 cards), I have 4 cards of Spades. So, other 3 players have total 9 cards of Spades. What is the probability that at least one player has no Spades?
- In the first move (Everyone has 13 cards), I have 4 Spades (3, 4, J, K). I want to play K, what is the probability of getting points (Q of Spades or any Hearts)?, remember that A of Spades > K of Spades.
Best Answer
Only answers your first question.
There are $39$ cards left for the other $3$ players and among them there are $9$ spades.
Each of them gets $13$ cards.
Number the other $3$ players and for $i=1,2,3$ let $E_i$ be the event that player $i$ gets no spades.
Applying the principle of inclusion/exclusion and symmetry we find that:$$P(E_1\cup E_2\cup E_3)=3P(E_1)-3P(E_1\cap E_2)=3\left[\frac{\binom{30}{13}}{\binom{39}{13}}-\frac{\binom{30}{4}}{\binom{39}{13}}\right]=$$$$\frac{3\left[\binom{30}{13}-\binom{30}{4}\right]}{\binom{39}{13}}\approx0.044223$$