Multiple hand "single event" probability blackjack

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Hi, does anyone know of any blackjack books that analyze conditional probability strategies when playing multiple hands such that the conditional probability determines optimal strategy for the multiple hands as a single event?

For example, assume two hands:

Hand 1: KK (two kings so total is 20)
Hand 2: K5 (one king, one five so total is 15)
And the dealer shows a 8 say ...

hand 1 has a high % chance to win
hand 2 has a much lower % chance to win

In this case, hand 2 is likely a loser, and hand 1 a winner, so perhaps optimally I take my win on hand 1 by standing, and cut my losses on hand 2 by surrendering. Thus, not playing the hands as separate probability events per 'Basic Strategy', but as a single conditional event (i.e. given that Hand 1 is likely a winner, and then using that probability to win for the probability determination for what to do on hand 2). I am angling for determining the optimal strategy for both hands as a single event.

I would like to find a book that does the conditional probability mathematical analysis on this.

Thank you.

Why should the optimal policy for Hand 2 be conditional on Hand 1?

It would be a personal preference to play. If one of the hands is likely a winner, you could use that information to mitigate risk on the other hand.

By surrendering the likely loser hand, mathematically, you would be "risk sharing" the risk from the loser hand to the winning hand (i.e., expecting the winner to make up for the loss of the surrender). The likely winner hand would need to have a positive expected return after it absorbs the loss of surrendering, or it would not make sense to do.

I have never heard of surrendering a hand before.

Do they do it for all players or only for the inexperienced?

Based on what I just read after googling the surrendering of hands in blackjack, I would respectfully disagree. I don't think you would surrender a hand just because the other hand was better. Each hand you are playing would need to be played or surrendered on its own strengths and weakness completely independent of the other hand.

There's not much difference between playing one hand and two hands.

In the situation you described, if you are playing with only one deck of cards, then you've got (I think) 6 cards of the 50 cards split across 3 hands - hand 1, hand 2, and the dealer. The next card drawn will be one of 44, and at this moment there are 45 unknown cards because the dealer's hand only shows 1 of its 2 cards.

So suppose you hit on hand 1 (you wouldn't since you'd almost certainly bust, but lets say you did). The only impact this has on hand 2 is that if you hit on hand 2, you are dealing with receiving one of 43 possible cards though at that moment there would be 44 unknown cards since the dealer still doesn't show one of his cards. Whatever card you drew for hand 1, you'd have less chance of getting that card if you hit on hand 2 (anywhere from 25-100% less chance), and slightly more chance of getting every other card that was not that one card.So, whether or not you hit on hand 1 has only minimal impact on what happens on hand 2. The impact is even less if you are playing with multiple decks of cards.

Likewise, what you do with hand 1 or 2 only has a minor impact on what happens to the dealer. If you hit on both hand 1 and hand 2, the dealer is now choosing one of 42 remaining cards, so the odds of him drawing the same card that you got in hand 1 or hand 2 go down, but the odds of him drawing any other card goes up slightly.

As I said , it is personal preference, so your statement of -- I don't think you would -- makes no sense.

Take this very simplistic example of only playing 2 hands one time ...

Assume equal bets:
Hand 1 is a blackjack and you win (pays 3:2)
Hand 2 is probabilistically a loser

You can surrender hand 2 and take a guaranteed total win of 50% gain on your total bet, or you can risk it, play out hand 2, and may end up winning only a 25% total gain on your total bet, if hand 2 happens to lose. This is the mathematics of minimizing your exposure to risk (i.e., "risk sharing"). You don't have to take risk on hand 2 because you are already a winner.

You don't appear to be addressing the mathematics of "risk sharing" which is what this thread is about.

Let's look at car insurance companies. Every car driver is different, like blackjack hands, however, the insurance companies don't view their risk that way, they share their risk between "good drivers" ("good hands") and "bad drivers" ("bad hands").

Your response does not seem to address this.

I'm pretty sure the optimal strategy for playing multiple hands as a single event is to play each hand individually based on probability.

I showed an example above where it is not

Repeated from above:

Assume equal bets:
Hand 1 is a blackjack and you win (pays 3:2)
Hand 2 is probabilistically a loser

You can surrender hand 2 and take a guaranteed total win of 50% gain on your total bet, or you can risk it, play out hand 2, and may end up winning only a 25% total gain on your total bet, if hand 2 happens to lose, and losing is the likely outcome.

Optimally it is better to take a 50% gain than to risk playing a statistically likely loser hand, and most likely end up with only a 25% gain.

Last edited by LoveNotHate on 24 Jul 2014, 8:12 pm, edited 1 time in total.

No, you've actually played both hands individually based on probability followed by a mathematical rationalization. Playing each hand individually based on probability will add up to best overall outcome. There is nothing more to it than that.

In my example, the second hand is surrendered before being played.



There are no guarantees when one takes risk. Have you ever been to a casino to see people lose miserably because they played as you say based on individual probability? The 'Basic Strategy' advice is to play many, many hands, however, not everyone wants to play that many to see the probabilities pan out. So, short of the million+ hands required, a risk sharing strategy can be used.

I showed above that one could take a 50% gain, and walk away, yet you propose to risk it, and likely end up with only a 25% gain. That is not the strategy I seek.

Yes I know, and that's the right thing to do.


Indeed. Risk is by definition an uncertainty implying a probability of loss. The odds are stacked in favour of the house: It will always win due to the law of large numbers. That's why you shouldn't gamble in the first place.


No, I've never proposed anything like that.

No one's saying that surrendering hand 2 is wrong. They're saying that surrendering it is correct independently of the state of hand 1. Surrendering the bad hand 2 would still be optimal even if hand 1 were also a bad hand, or even if hand 2 was the only hand you had.

EDIT: When you have a small amount of money, low risk strategies can be better than higher risk strategies even if the latter have greater expected value. When you have a large amount of money, maximizing expected value is optimal. Mathematics could be used to attempt to derive an optimal strategy for any given combination of goals and resources.

Last edited by Tiranasta on 25 Jul 2014, 4:52 am, edited 1 time in total.

True, but I think we're dealing with a semantic confusion related to whether a surrendered hand is technically considered played or not.

'Basic blackjack strategy' is more or less the optimal strategy for playing hands individually, and 'Basic blackjack strategy" says it is wrong.

Assume this scenario:
Dealer: 8
Your Hand 1: Blackjack (pays 3:2)
Your Hand 2: King and a four

Basic strategy (i.e., "playing the hands individually") says to hit on the likely losing hand 2, whereas, I am suggesting to surrender the second hand and take the 50% gain.



'Basic Strategy' has you hit or stand on hands where you have a high % probability to lose. So, no it would not be optimal per 'Basic Strategy' to surrender a hand just because it was likely a losing hand.

The present optimal decision is reached by looking at winning/likely-to-win hand and decide based on that probability what to do on the second hand.

Last edited by LoveNotHate on 25 Jul 2014, 5:11 am, edited 1 time in total.