Probabilities

[anthony coussa]

I would like to find out the following:

In the game of "Blackjack" or "21" what are the odds of making a hand, 17 and also 14, using a single standard deck of 52 cards?
e.g., 10+7;* 11(ace)+6;* 8+6+3; 9+8; etc.
10+4;* 9+5; 8+6;* 7+7; 10+2+2; etc.

P.S: What would the difference be in the results or answers, if six or eight decks are used instead of a single deck?

Your help will be greatly appreciated. Or, where can I go on the web to find out?

Anthony

 

[KenSmith]

First you'll need to specify a strategy of a hard total and soft total where you would stop hitting. Then you can calculate the probability of each ending hand.

If you choose one of two specific strategies, the results are available online.

The strategy "h17s17" hit to hard 17 or soft 17, whichever comes first, is identical to the dealer strategy when they stand on all 17s.

The strategy "h17s18" hit to hard 17 or soft 18, whichever comes first is the dealer strategy in an "H17" game.

Those resulting hand probabilities are available here, for various numbers of decks:
Dealer Outcome Probabilities

For other strategies, you'll likely have to calculate them yourself, although we occasionally discuss such things over at https://www.blackjacktournaments.com/

 

[ThunderWalk]

I would like to find out the following:

In the game of "Blackjack" or "21" what are the odds of making a hand, 17 and also 14, using a single standard deck of 52 cards?
e.g., 10+7;* 11(ace)+6;* 8+6+3; 9+8; etc.
10+4;* 9+5; 8+6;* 7+7; 10+2+2; etc.

P.S: What would the difference be in the results or answers, if six or eight decks are used instead of a single deck?

Your help will be greatly appreciated. Or, where can I go on the web to find out?

Anthony

A little Google work might answer some of your questions.

(Dead link: http://www.bettingrevue.com/games/blackjack/blackjack-odds.htm)

Odds for Single Deck Blackjack
- Dealer stands on soft 17 -0.0013%
- Dealer hits on soft 17 0.1896%

Odds for Double Deck Blackjack
- Dealer stands on soft 17 0.3359%
- Dealer hits on soft 17 0.5374%

Odds for Four deck blackjack
- Dealer stands on soft 17 0.4953%
- Dealer hits on soft 17 0.7128%

Odds for Six deck blackjack
- Dealer stands on soft 17 0.5479%
- Dealer hits on soft 17 0.7672%

Odds for Eight deck blackjack
- Dealer stands on soft 17 0.5742%
- Dealer hits on soft 17 0.7945%

 

[KenSmith]

That's not what he's asking. He's looking for the probabilities of getting to specific totals, and the numbers you provided are EV for the game with various rules.

 

[jimpenn]

Imagination is most important then knowledge. For while knowledge defines all we currently know and understand, imagination points to all we might yet discover and create. - Albert Einstein

 

[anthony coussa]

That's not what he's asking. He's looking for the probabilities of getting to specific totals, and the numbers you provided are EV for the game with various rules.

The above is absolutely correct. As an example, we know in a dice game,
the odds of rolling two 6's or two 1's: 1 in 36 or, 35 to 1 should be the payoff
but the house only pays 30 to 1 and this is how they have the advantage, in the long run.

In Roulette, since there are 36 numbers plus the 0 and 00, the probability is
obviously 1 in 38 or, 37 to 1 should be the payoff but the casino pays only 35 to 1, again, resulting in the casino's built-in advantage.

Using these examples, back to my question of what are the odds or probabilities (with 2 or more cards) of having a "17 hand" or a hand of "14", or even, 15, 16, for that matter. Assuming six or eight decks are used. The decision to draw or not to draw to such hands is really irrelevant. Do I need a mathematician to figure this out? Or, is there in existence a table showing the odds of having ANY hand in the game of Blackjack/21?

Anthony Coussa

 

[shadroch]

Wouldn't you only be able to compute the odds of a fresh shoe? Once a single card comes out,you wouldn't get the same results.
The examples you mention are independant trials,where the results of the last play has no effect on the new play. BJ is very dependent on the previous hands.For example-If all four Aces come out in the first round of a single deck game,you know that no one will get a BJ the rest of the deck.
Would you mind answering why you think this information is important?I,for one,don't see how it is.

 

[ColorMeUp]

Using these examples, back to my question of what are the odds or probabilities (with 2 or more cards) of having a "17 hand" or a hand of "14", or even, 15, 16, for that matter. Assuming six or eight decks are used. The decision to draw or not to draw to such hands is really irrelevant. Do I need a mathematician to figure this out? Or, is there in existence a table showing the odds of having ANY hand in the game of Blackjack/21?

Anthony Coussa

What you have to do is figure out all the ways to make 17 (e.g. 10+7, 11+6, 9+8, 8+8+1, etc), calculate the probability of making the hand for each (e.g. for 10+7 it would be 4/13*1/13*2 [times 2 since you can make it with 10 then 7 or 7 then 10, you can also do it with a combination]) then sum the probabilities for all ways to make the hand you want.

 

[KenSmith]

The decision to draw or not to draw to such hands is really irrelevant. Do I need a mathematician to figure this out? Or, is there in existence a table showing the odds of having ANY hand in the game of Blackjack/21?

Are you asking just about the probability of the initial two-card deal? Because if not, then the decision on whether to draw or not draw is quite important. For example, the probability of getting a 21 probably shouldn't include the hand (T,T,Ace). But, if you were in a tournament situation where standing wouldn't win, and you didn't have enough money to split, hitting 20 might make sense.

Now, if you just want initial two-card hand probabilities, those are easy to calculate. The others take more effort.

 

[anthony coussa]

Hi Ken,

Before I respond I wish to express my sincere appreciation for your interest
and very prompt response.

1- The answer to your first question is yes, naturally as long as they add up to 12, 13, 14, 15, 16 and up to 20 inclusive.

2- A seperate calculation COULD BE, to draw to 12 through 16, of course when the dealer shows 7 and a higher value card and naturally, stand on these hands when the dealer shows 6, 5, 4, 3 and 2.

Again, thank you so much for your help.

Anthony

 

[KenSmith]

Probability of initial two-card hand totals, 6 decks:

Hard 4: 0.005688845
Hard 5: 0.011872372
Hard 6: 0.017561217
Hard 7: 0.023744744
Hard 8: 0.029433589
Hard 9: 0.035617116
Hard 10: 0.041305961
Hard 11: 0.047489488
Hard 12: 0.088795449
Hard 13: 0.083106604
Hard 14: 0.076923077
Hard 15: 0.071234232
Hard 16: 0.065050705
Hard 17: 0.05936186
Hard 18: 0.053178333
Hard 19: 0.047489488
Hard 20: 0.093989612

Soft 12: 0.005688845
Soft 13: 0.011872372
Soft 14: 0.011872372
Soft 15: 0.011872372
Soft 16: 0.011872372
Soft 17: 0.011872372
Soft 18: 0.011872372
Soft 19: 0.011872372
Soft 20: 0.011872372

Blackjack: 0.047489488

 

[QFIT]

Certainly close enough. But those numbers ignore the cut-card effect.

 

[ColorMeUp]

Hi Ken,

Before I respond I wish to express my sincere appreciation for your interest
and very prompt response.

1- The answer to your first question is yes, naturally as long as they add up to 12, 13, 14, 15, 16 and up to 20 inclusive.

2- A seperate calculation COULD BE, to draw to 12 through 16, of course when the dealer shows 7 and a higher value card and naturally, stand on these hands when the dealer shows 6, 5, 4, 3 and 2.

Again, thank you so much for your help.

Anthony

That calculation is easy if you do it the way I suggested. You would just not include any of the hands where you wouldn't hit.

 

[KenSmith]

Yes, these numbers are "off the top" of a shuffled 6 deck shoe.

 

[EasyRhino]

Certainly close enough. But those numbers ignore the cut-card effect.

What does the cut card effect mean again? Is this the same as card removal effect (i.e if you pull one 9 out of the deck, then that ever so slightly reduces the chances of getting a second nine)? Or something else?

 

[Canceler]

What does the cut card effect mean again?

If a lot of small cards come out early, the count rises. But since they're small cards, more of them will be used, because the players will take more hits. This will result in reaching the end of the shoe (the cut card) sooner, with a high count.

Conversely, if high cards come out early, the count will drop, and the shoe will last longer.

None of this is anything to lose sleep over.