Can anyone confirm expectations for the game of blackjack?

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...

Solution 1:

Not an answer, but too long for a comment.

In order to verify these calculations, we need to know what assumptions you're making. For example, a lot of blackjack analysis assumes an infinite deck; in other words the cards are being drawn from the deck "with replacement".

If you're assuming an finite deck, then we'd need to know how many decks are in your shoe.

Although the Dealer's play is completely rigid, you need to specify exactly what decision tables your Player is following. Unless you're computing the optimum strategy on the fly?

You might be using the standard tables, but even those are merely approximations. For example, you should, in general, hit 16 against a 10. But what if you're holding 2 2 2 3 3 4? I'm not so sure you want to hit that hand with so many small cards missing from the deck.