Basic Strategy

Hey guys,

I posted a few days ago asking for some sources to verify the expected returns you have depending on your card and the one of the dealer. These expected return are what helps you establish basic strategy. And re verifying basic strategy is my goal !! So, I was redirected to the wizard of odds appendix: /games/blackjack/appendix/1/.

I verified what I had found when you stand with a 21. I have the same results as the wizard except for the 10 and ace and I don't understand the wizard's results. Let me explain:
- For exemple when you stand with a 21 and the dealer has an ace, the wizard says you have 0.922194 expected return. But let's simplify our experience and say that except when the dealer has a 10, you win and double your bet. We are simplifying and what we will find will be a slightly bigger expected return than in reality (because the dealer can tie with you by getting other cards for example two 5).
And yet we find that the average gain (if you bet 1) is: (16/52)*1 [representing the case when there is a tie]+ (36/52)*2 [when you win and double your bet] =1.69230769231
We have a 0.69230769231 average gain.
As you can see even though we should get a higher than what is shown in the appendix, we get a significantly lower expected return. Could someone tell me if I made a mistake or if there is a problem with the appendix.
Thanks for reading,
Julian

A two card 21 is a blackjack and returns 2.5x (gains 1.5x).
Edit: Also a player blackjack ties a dealer two-card 21, but is payed in full even if the dealer draws to 21. That should make your calculation easier.

Quote: JimRockford

A two card 21 is a blackjack and returns 2.5x (gains 1.5x).
Edit: Also a player blackjack ties a dealer two-card 21, but is payed in full even if the dealer draws to 21. That should make your calculation easier.

But in the appendix( wizardofodds . com/games/blackjack/appendix/1/ ) he's talking about standing on 21, isn't it implied that it is not a blackjack ? This could explain the difference, but how did he calculate the expected return if without differenciating the two cases (blackjack and 21 made of more than two cards) ?
Especially since, in the other cases (when the dealer has from 2 to 9) I have the exact same expected returns (without considering the 21 as a blackjack).

I think the wizard's number when the dealer has an Ace up-card is after Dealer has checked for Blackjack. Once you know the dealer hasn't got a 10 as the down-card, the chances of Dealer making 21 (not BJ) is actually quite low. The table below Stand soft 17 (/ Hit soft 17) confirms that the dealer will get 21 7.78% of the time, so the other 92.22% time you'll win 1 unit if you have 21.

21	7.78%	9.57%
20	18.89%	20.69%
19	18.89%	20.69%
18	18.89%	20.69%
17	18.89%	8.30%
Bust	16.65%	20.07%

I've typed these numbers in, so please forgive me if any typos.
Note: These figures do not apply to European BJ where the dealer does not take a second card until after all Players have acted.

Quote: charliepatrick

I think the wizard's number when the dealer has an Ace up-card is after Dealer has checked for Blackjack. Once you know the dealer hasn't got a 10 as the down-card, the chances of Dealer making 21 (not BJ) is actually quite low. The table below Stand soft 17 (/ Hit soft 17) confirms that the dealer will get 21 7.78% of the time, so the other 92.22% time you'll win 1 unit if you have 21.

21	7.78%	9.57%
20	18.89%	20.69%
19	18.89%	20.69%
18	18.89%	20.69%
17	18.89%	8.30%
Bust	16.65%	20.07%

I've typed these numbers in, so please forgive me if any typos.
Note: These figures do not apply to European BJ where the dealer does not take a second card until after all Players have acted.

Hey charliepatrick,

Thanks I didn't think about this ! I like your answer, I will have to check in detail but I don't see any other explanation. I still have a question, why do these figures don't apply to European BJ, what difference does it make to take the card at the beginning or after all Players have acted ?

Julian

In the US your [three+ card] 21 cannot lose to a BJ as the dealer has already checked. In Europe your 21 could still lose to a BJ, so your chances of winning are much less. Technically all the percentages are multiplied by 9/13 and a row with 4/13 dealer gets BlackJack added. So your 21 only wins 9/13*92%, i.e. about 63.84%.