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I computed the probabilities and expectations for blackjack in Mathematica and here is what I came up with:

If player is given only the option to stand the expected win per unit bet is

$$ -\frac{631462897715505}{3937376385699289}. $$

If player is now allowed to hit the expectation is

$$ -\frac{2220691644539301303808782629}{91733330193268616658399616009} $$

If further player is allowed to double we get

$$ -\frac{1071160328643044865131012313}{91733330193268616658399616009} $$

and if splitting is also an option we get

$$ -\frac{3305320393577010676623056559}{1192533292512492016559195008117} $$

There is a reason I posted those values with infinite precision. I modeled the game and used basic probability on the states of this system to calculate them. No simulations, no Monte Carlo, just pure math and careful numbering. Has anyone done something similar to confirm or disprove me ? An internet search didn't reveal much by the way...

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  • $\begingroup$ What rules are the dealer following in each case? $\endgroup$ – Ted Nov 5 '18 at 16:34
  • $\begingroup$ In all cases the dealer plays the same, like a computer: taking card if the score of his hand is less than 17 and stopping if greater or equal to 17. These are standard rules. $\endgroup$ – plus1 Nov 5 '18 at 19:07

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