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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.
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  • $\begingroup$ what have you tried to solve this problem? $\endgroup$ – supinf Apr 5 at 14:23
  • $\begingroup$ How many ways are there that every player has a spade? Isn't it the partition of $9$ into $3$ parts. Which is same as the number of surjective maps from a $9$ elements set to a $3$ elements set. $\endgroup$ – Dbchatto67 Apr 5 at 14:37
  • $\begingroup$ I'm not good in probabilities & statistics. I thought for the first problem, no one has spades probability of 0, one player has spades probability of 1/9, two player have probability of 1/81, everyone have spades probability of 1/3. is it correct? @supinf $\endgroup$ – Tareq Joy Apr 5 at 14:42
  • $\begingroup$ @Dbchatto67 can you explain me more, please? $\endgroup$ – Tareq Joy Apr 5 at 14:56
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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:$$P(E_1\cup E_2\cup E_3)=3P(E_1)-3P(E_1\cap E_2)=3\left[\frac{\binom{30}{13}}{\binom{39}{13}}-\frac{\binom{30}{4}}{\binom{39}{13}}\right]=$$$$\frac{3\left[\binom{30}{13}-\binom{30}{4}\right]}{\binom{39}{13}}\approx0.044223$$

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  • $\begingroup$ Isn't 0.044223 the probability that no one gets Spades? I wanted at least one player who doesn't get Spades! $\endgroup$ – Tareq Joy Apr 5 at 16:16
  • $\begingroup$ So, I subtract it from 1? 1-0.044223? $\endgroup$ – Tareq Joy Apr 5 at 16:24
  • $\begingroup$ $0.044223$ is the probability that at least one of the $3$ has no spades. $\endgroup$ – drhab Apr 5 at 16:29
  • $\begingroup$ thanks, can you solve the second one please? I couldn't continue my game developing! :( $\endgroup$ – Tareq Joy Apr 5 at 16:49
  • $\begingroup$ Sorry, but for that I must at least know the rules of the game. I am not a cardplayer and have absolutely no idea how it works. Your second question on its own does not make me wiser. Btw, don't interpret this as promise. $\endgroup$ – drhab Apr 6 at 9:21

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