Unconditional Expected Winnings
I wish to calculate the expected winnings for a strategy that I have developed (not optimal) based on only the player's knowledge of the dealer's upcard and also the unconditional expected winnings (House edge). I have already calculated exactly the expected winnings to the player for each player hand vs each House upcard. However I would like to extend the EV's that I have already calculated to the two expectations I have outlined above and would really appreciate any help on how to go about this.
Many thanks.
Quote: MangoJIf you know the EV for given player hand and upcard, and wish to know the EV for a given upcard alone, simply sum over all player hands with their corresponding probability (of course respecting the dealers upcard). To get to the unconditional EV, simply sum over all upcards (again with their corresponding probability).
Thanks that's great, this is kind of as I thought but i was trying to avoid what I imagine is the the mammoth task of calculating the probability of a given player hand. How would i go about this? My state space for the player's hands are Bust, Blackjack, 21:12, 11N:4N (N refers to no ace) 11A:2A (where A refers to Ace in hand valued as 1). I realise I can write out all the different ways of getting the hands and the probabilities of getting those cards but there must surely be a quicker way. Could anyone link me to any similar examples of how to do this?
If your players space does not include card removal effects, you are effectively doing an infinite deck analysis which is usually pretty easy to do (could be done in a spreadsheet). There are multiple ways of an infinite deck analysis. In the simplest case just write down the transition matrix between your hands, and solve it consistently.
As a remark: you need to distinguish between "soft" and "hard" hands. A soft hand is a hand containing an Ace currently counting as 11 (with the option of counting it as 1). A "hard" hand is a hand, where there are either no Aces present, or if there are Aces all must be counted as 1.
So your hand space should be: Bust, BJ, hard 21 (no BJ), hard 20-12, soft 20-12, 11-4. The game mechanics will ensure that all hands (expect splits) evolve in the same direction, making your transition matrix a lower (or upper) diagonal.
Quote: MangoJWhen you do exact EV calculations in Blackjack, that is what you need to do: traverse through all possible ways getting each hand, and sum up their EV against the upcard. The actual traverse makes sure that all card removal effects (well, basically except for splits) are respected.
If your players space does not include card removal effects, you are effectively doing an infinite deck analysis which is usually pretty easy to do (could be done in a spreadsheet). There are multiple ways of an infinite deck analysis. In the simplest case just write down the transition matrix between your hands, and solve it consistently.
As a remark: you need to distinguish between "soft" and "hard" hands. A soft hand is a hand containing an Ace currently counting as 11 (with the option of counting it as 1). A "hard" hand is a hand, where there are either no Aces present, or if there are Aces all must be counted as 1.
So your hand space should be: Bust, BJ, hard 21 (no BJ), hard 20-12, soft 20-12, 11-4. The game mechanics will ensure that all hands (expect splits) evolve in the same direction, making your transition matrix a lower (or upper) diagonal.
Thankyou very much!
