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So when I look online for this everything wants to talk about just BlackJacks themselves but I want to find the probability of any player getting $21$ after having hitting $N$-times when playing vs a casino with $X$-deck shoes. For simplicity I only need to know the values with $n$ is $1, 2$, or $3$ and when $X$ is $6, 7$ or $8$. But a generalized formula would be great.

For a baseline we can assume just the player vs dealer but if having more players affects the odds of any player getting this I would be intrigued to know how to account for that. Max players of $7$.

If this is easier to do with code instead of formulas that works for me as well.

Thanks for any help you can provide.

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  • $\begingroup$ More players does not affect this at all. Knowledge of other players' cards however does. This is a highly tedious question with a lot of case work. You can do better by considering an infinite shoe instead and using that as an approximation. To do so, look at the $x^{21}$ coefficient of the expansion of $(x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+4x^{10}+x^{11})^{n+2}/13^{n+2}$ noting that this will be a slight overestimation, not only because of the infinite shoe but also because we allowed aces to appear twice in the above in both respective roles. $\endgroup$
    – JMoravitz
    Sep 1, 2020 at 22:29
  • $\begingroup$ That said, brute force should be feasible... We look at all partitions of $21$ into $n$ parts, all of size $11$ or less, there are only so many. Looking at partitions instead of composition helps reduce the number of cases. We have $49$ different cases for $n=4$. For each case, decide which deck and suits are used for each respective rank (as well as which face card in the case of 10's). For instance for the $11+4+4+2$ case there are $24\times \binom{24}{2}\times 24$ different four-card hands (order not mattering) out of the $\binom{52\times 6}{4}$ total possible hands in this case. $\endgroup$
    – JMoravitz
    Sep 1, 2020 at 22:39
  • $\begingroup$ @JMoravitz thanks for these comments.I will do some googling to see if I can learn how to use your suggestions. $\endgroup$ Sep 10, 2020 at 16:09

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