Confused

My math skills are limited. But I seem to have found some flaws in the basic strategy for Blackjack, maybe? To simplify, take the hand A,2 with a 6 up. The analyzer on the Wizard's simulator recommends doubling down and provides the following analysis: a double down provides a return, per dollar bet, of 0.230210. As compared to a lessor return per dollar bet when hitting: 0.168495. But because the option of hitting allows more wins overall, due to the options of additional hits, all of which are risk-free, for example an A,2 can draw an Ace, a 2, 3, 4, or even a 5, and still take another hit without any risk of busting. So, if this hand is played in accordance with Basic Strategy, a player will never bust and win much more often when hitting rather than when doubling down (of course doubling down only allows a draw of one card). So, on a per dollar bet basis, how might the return when doubling down be higher than when hitting?

I will have to look at this some more, but I have a feeling the answer is, "per dollar bet" refers to only the original bet - not the additional bet when you double.
In other words, if you bet $100 and are dealt A,2 against a dealer's 6:
If you hit, you are expected to make a profit of $16.85.
If you double, you are expected to make a profit of $23.02.

As far as basic strategy:

Over time, if you double when you should, you should win more money.

Just as, if you split 8s against 10, over time, you should lose less.

Perhaps that is correct, it makes sense. But the simulator's recommendations would be more telling if the comparison between hitting or doubling were giving the actual difference in money gained or lost per choice. Naturally, there are other hands which have this same issue, mostly the other soft hands such as A,3 - A,4 - and so on, but then too there is the hard 8 and 9 which will win more hands when hit as opposed to doubled (assuming here again that the Basic Strategy is adhered to).

Thanks for the response, just trying to better understand the game.

Essentially, in general, by doubling you are sacrificing the opportunity to win when you would want to draw another card (e.g. A22) compared with winning twice as much when you do win. So typically you will win less often by doubling, but because you win two units rather than one unit, it's more profitable.
(Usually you double where your chances of winning is greater than 50%, but technically there are several exceptions for some Blackjack variants like FreeBet or where the initial/doubled bet pays more than evens.)

Thanks for the response. And yes, doubling a bet will double the winnings but it will also double the losses. So, on a per dollar bet basis, how can winning 53 hands out of a hundred with a $10 bet (doubling down), have a better return per dollar bet if by hitting with a $5 bet wins more than 53 hands out of a hundred? Half as much was bet, but more was won per dollar bet.

(Assume for simplicity that there aren't ties, or if you want to be more specific that if you double you win 6% more often than losing, etc.)
Case (a) You double and have $10 out winning 53% and losing 47%. Net profit is 6% * $10.
Case (b) You hit and have $5 out winning 55% and losing 45%. Net profit is 10% & $5.
As you can see 6*10 is greater than 10*5.

If you look at doubling 10 vs 9 ( https://wizardofodds.com/games/blackjack/expected-return-infinite-deck/ ) you get 14c or 11c, which suggests the numbers are 7% (dbl) and 11% (hit).

Quote: RLL123

Thanks for the response. And yes, doubling a bet will double the winnings but it will also double the losses. So, on a per dollar bet basis, how can winning 53 hands out of a hundred with a $10 bet (doubling down), have a better return per dollar bet if by hitting with a $5 bet wins more than 53 hands out of a hundred? Half as much was bet, but more was won per dollar bet.
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You want more money on the table when you have your best opportunities for a win. A/2 vs 6 is a strong hand, and it is worth giving up a small winning percentage to double your win. That assumes you have a proper bankroll available.

Still, that time I doubled on beyond table limit on an ace 6 against 3 up, and lost, does stand out. In that particular hand I drew a 5, and dealer turned over an 8 under the 3 and drew a 9. I will hit hard 12 against 3 up one time absent some other knowledge of what might be coming, so that makes the recollection sharper.

But there are plenty of hands where I doubled with massive amounts down and won. More than I have lost, for sure.

Jack King said, "Few players recall big pots they have won, strange as it seems, but every player can remember with remarkable accuracy the outstanding tough beats of his career."

Yes, Charliepatrick, I understand that "you are sacrificing the opportunity to win when you would want to draw another card (e.g. A22) compared with winning twice as much when you do win. So typically you will win less often by doubling, but because you win two units rather than one unit, it's more profitable."

But it is not more profitable 'per dollar bet'. This is however somewhat confusing, but if a player were to win 58 times out of 100 hands at $1 per hand, and lose 42 times, when hitting, that player's profit would be $16, or 0.16 per dollar bet. In comparison, a player doubling down and betting $2 per hand a hundred times and winning the same number of times, 58, would win twice as much, naturally, but the same on a per dollar bet basis. Twice as much bet, and twice as much won, yet via the analyzer on 'simulator 2', and in the appendix here on this site, the analysis has the double down option winning more per dollar bet (Double Down @ 0.230210 - Hit @ 0.168495).

Thanks for the interest.

A good point billryan, but still not explaining how an A,2 could possibly return more 'per dollar bet' when doubled on when the hit play will win more hands overall. Maybe there is something I'm unaware of, but isn't the Basic Strategy based on a 'per dollar bet' analysis, combinatorial or otherwise?

Quote: RLL123

A good point billryan, but still not explaining how an A,2 could possibly return more 'per dollar bet' when doubled on when the hit play will win more hands overall. Maybe there is something I'm unaware of, but isn't the Basic Strategy based on a 'per dollar bet' analysis, combinatorial or otherwise?
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It’s on an EV basis per dollar of base bet. (So doesn’t include the added money for doubling or splitting.)

Thanks, unJon, your explanation supports what 'ThatDonGuy' suggested in the first response. It seems odd though considering how meticulous math-nurds are about such things that so many have ignored what is clearly misleading. The choice is 'to hit, or not to hit'... or to risk more due to the presented opportunity... and double down, or not? So, in trying to make an informed decision, it would be better to know which option is the better play... per dollar bet.

Besides that, the listed probabilities are presented as evaluations of wins and losses 'per dollar bet'.

And of course, more profit is predictable with twice the bet on a hand with a win rate of better than 50%, so the numerical comparison offered is not as useful as an actual comparison of a 'return on investment' calculation.

As a player, you will play a finite amount of hands. Having bigger bets on strong hands will result in more money when you stop playing. Isn't that what you want?

So are you unconfused now?

Quote: RLL123

Thanks, unJon, your explanation supports what 'ThatDonGuy' suggested in the first response. It seems odd though considering how meticulous math-nurds are about such things that so many have ignored what is clearly misleading. The choice is 'to hit, or not to hit'... or to risk more due to the presented opportunity... and double down, or not? So, in trying to make an informed decision, it would be better to know which option is the better play... per dollar bet.

Besides that, the listed probabilities are presented as evaluations of wins and losses 'per dollar bet'.

And of course, more profit is predictable with twice the bet on a hand with a win rate of better than 50%, so the numerical comparison offered is not as useful as an actual comparison of a 'return on investment' calculation.
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APs use metrics like you are describing all the time. We use risk-averse indices, because a play like A2 vs. 6 is going to be more or less profitable depending on the count. That's a Basic Strategy play so it is profitable right off the top of the deck.

What you're saying is technically correct, but practically not useful. If you're a counter you are betting a tiny fraction of your bankroll on any hand, so whether or not you double isn't going to break the bank, and making your indices risk averse doesn't change them much. The increased win rate you get for doubling is going to be well above the expected win rate for putting out that bet at that count in the first place; if you expected 1% when you placed the bet getting 1% on the doubled bet is all you're looking for, and that's going to be the case for all doubled hands except in rare circumstances.

As others have said, the expected value is relative to the initial bet. Yes, the probability of winning is more by hitting, but by doubling you get more action on a good hand.

What would you rather have?

A) 80% chance of winning $1, otherwise lose $1
B) 70% chance of winning $2 otherwise lose $2.