Blackjack card count probability

Hey there,
Need to start with the fact that I just love your work! I know game providers that are implementing new games and they are basing their mathematics on your calculation (more likely copying your calculations).

I work in a gambling domain, and I do all kind of statistics on historical data in order to see if occurrence rates for various scenarios are in the right "range".
When it comes to theoretical mathematic, I'm sad to say I have a long road ahead :(

I have a question that mess with my mind and on my own I can't get to the bottom of it unless I put in some serious hours of work based on historical data, but the main idea is this:
How can I calculate on Blackjack 8 deck ( 6 out of 8 decks dealt) the percentage of hands won in the scenario that someone is playing only when the true count of a shoe is = or > than 1?

Thank you for everything you do, keep it you and need to mention I will still love your work even if you won't find the time to answer me.
All the best!

Interesting question! I hope Wizard see this, but it’s probably very hard to calculate the win rate because it depends on specific rules. Let me just ask a similar but simpler question.

For an 8-deck shoe game, the probability of a blackjack is 4.745%. What is the probability of a player’s blackjack when player only plays at true count >=+1?

Quote: aceside

Interesting question! I hope Wizard see this, but it’s probably very hard to calculate the win rate because it depends on specific rules. Let me just ask a similar but simpler question.

For an 8-deck shoe game, the probability of a blackjack is 4.745%. What is the probability of a player’s blackjack when player only plays at true count >=+1?
link to original post

Some quick simulating using Hi-Lo gets around 5.435%

I only have six deck (UK rules) at 66% penetration and count winning BJs (as others are counted in the "Ties"). The base number was 16795488 (4.528%) winning BJs in 370930867 hands (100m shoes). There were 92045368 hands with count 1 or greater and 4670271 of those were winning BJs (5.074%)

Quote: ThatDonGuy

Quote: aceside

Interesting question! I hope Wizard see this, but it’s probably very hard to calculate the win rate because it depends on specific rules. Let me just ask a similar but simpler question.

For an 8-deck shoe game, the probability of a blackjack is 4.745%. What is the probability of a player’s blackjack when player only plays at true count >=+1?
link to original post

Some quick simulating using Hi-Lo gets around 5.435%
link to original post

Let me use your number to calculate the player's edge from additional blackjacks.
5.435%x(1-5.435%)x1.5-4.745%x(1-4.745%)x1.5=0.9296%.

This means a HiLo counter will be able to obtain a 0.9296% edge over the house by wonging in at the true count of greater than or equal to +1, betting one unit flat all the way. It is easy to overcome the house edge, isn't it?

Quote: aceside

Quote: ThatDonGuy

Quote: aceside

Interesting question! I hope Wizard see this, but it’s probably very hard to calculate the win rate because it depends on specific rules. Let me just ask a similar but simpler question.

For an 8-deck shoe game, the probability of a blackjack is 4.745%. What is the probability of a player’s blackjack when player only plays at true count >=+1?
link to original post

Some quick simulating using Hi-Lo gets around 5.435%
link to original post

Let me use your number to calculate the player's edge from additional blackjacks.
5.435%x(1-5.435%)x1.5-4.745%x(1-4.745%)x1.5=0.9296%.

This means a HiLo counter will be able to obtain a 0.9296% edge over the house by wonging in at the true count of greater than or equal to +1, betting one unit flat all the way. It is easy to overcome the house edge, isn't it?
link to original post

That's the (estimated) probability of you getting a blackjack. You're forgetting that the dealer is also more likely to get a blackjack.

I'll leave you to do the maths but at +1 there's only a minor advantage to the player. But you are correct wonging at +1 or more gives an advantage. On average (based on these sim figures and UK rules) it's 0.69% HE (advantage) and makes $18.36 profit per 100 hands watched for $100 bets. This is because the edge at lower counts is small and the higher opportunities don't come as often.
Here's an extract from the figures, note the sim works out count into 0.1 chunks so these are the values at 0,1,2 etc. The overall HE for the game is -0.48% (to the house).

Quote: ThatDonGuy

Quote: aceside

Quote: ThatDonGuy

Quote: aceside

For an 8-deck shoe game, the probability of a blackjack is 4.745%. What is the probability of a player’s blackjack when player only plays at true count >=+1?

Some quick simulating using Hi-Lo gets around 5.435%

Let me use your number to calculate the player's edge from additional blackjacks.
5.435%x(1-5.435%)x1.5-4.745%x(1-4.745%)x1.5=0.9296%.

This means a HiLo counter will be able to obtain a 0.9296% edge over the house by wonging in at the true count of greater than or equal to +1, betting one unit flat all the way. It is easy to overcome the house edge, isn't it?

That's the (estimated) probability of you getting a blackjack. You're forgetting that the dealer is also more likely to get a blackjack.

The (simulated) probability of getting a blackjack and the dealer not also getting a blackjack is 5.153%.

Quote: charliepatrick

I'll leave you to do the maths but at +1 there's only a minor advantage to the player. But you are correct wonging at +1 or more gives an advantage. On average (based on these sim figures and UK rules) it's 0.69% HE (advantage) and makes $18.36 profit per 100 hands watched for $100 bets. This is because the edge at lower counts is small and the higher opportunities don't come as often.
Here's an extract from the figures, note the sim works out count into 0.1 chunks so these are the values at 0,1,2 etc. The overall HE for the game is -0.48% (to the house).

Count: 0 Exp: -0.004362165317904683 Hands: 34058888 Win: 15389403 Lose: 17851740 Tie: 2998380 CHY: 0 BJk: 1542511
Count: 1 Exp: 0.0005946421593759342 Hands: 5576463 Win: 2502519 Lose: 2896196 Tie: 494838 CHY: 0 BJk: 264662
Count: 2 Exp: 0.0062002146315115495 Hands: 3095538 Win: 1381105 Lose: 1591982 Tie: 278383 CHY: 0 BJk: 153380
Count: 3 Exp: 0.009922477361631772 Hands: 1570122 Win: 695248 Lose: 801953 Tie: 142112 CHY: 0 BJk: 81523
Overall Result: Exp: -0.004796346592530947 Hands: 370930867 Win: 167589328 Lose: 194561673 Tie: 32670420 CHY: 0 BJk: 16795488

link to original post

Hands watched or hands played?

Here is the six deck analysis:

https://www.888casino.com/blog/blackjack-tips/blackjack-card-counting-vs-blackjack-side-bets-six-decks

Quote: ThatDonGuy

Quote: aceside

Quote: ThatDonGuy

Quote: aceside

Interesting question! I hope Wizard see this, but it’s probably very hard to calculate the win rate because it depends on specific rules. Let me just ask a similar but simpler question.

For an 8-deck shoe game, the probability of a blackjack is 4.745%. What is the probability of a player’s blackjack when player only plays at true count >=+1?
link to original post

Some quick simulating using Hi-Lo gets around 5.435%
link to original post

Let me use your number to calculate the player's edge from additional blackjacks.
5.435%x(1-5.435%)x1.5-4.745%x(1-4.745%)x1.5=0.9296%.

This means a HiLo counter will be able to obtain a 0.9296% edge over the house by wonging in at the true count of greater than or equal to +1, betting one unit flat all the way. It is easy to overcome the house edge, isn't it?
link to original post

That's the (estimated) probability of you getting a blackjack. You're forgetting that the dealer is also more likely to get a blackjack.
link to original post

You are right. Let me revise my calculation by taking into account the edge gained by the dealer too.
5.435%x(1-5.435%)x(1.5-1)-4.745%x(1-4.745%)x(1.5-1)=0.3099%.

This means that the AP player’s edge due to additional blackjacks from wonging is not enough to overcome the house edge of not counting. However, this portion is important, because it’s about 0.3099%/1.16%=27% of the total wonging advantage while flat betting (this 1.16% number is taken from Eliot’s publication in the above post). This 27% number looks good, but Wizard put this number as 7%. Discrepancy?

Quote: eqelolita

...
I work in a gambling domain, and I do all kind of statistics on historical data in order to see if occurrence rates for various scenarios are in the right "range".
When it comes to theoretical mathematic, I'm sad to say I have a long road ahead :(

...How can I calculate on Blackjack 8 deck ( 6 out of 8 decks dealt)...

A couple things to note here:

First, Good luck. You seem like a nice enough person but unfortunately many casinos (and I'm sure your own) have treated people also just trying to use their brain in blackjack (like you) very poorly, such as violating local and federal law to kidnap, detain, illegally obtain personal information, arrest, etc. Thus, most people don't want to help anyone or anything in the industry.

Next, I find it quite comical that the casino would hire a "math guy" to make sure their customers are in the "right range" also whilst shooting themselves in the foot dealing only 6 out of 8 decks. What you should do is the math on how much more money the casino would make if every single table dealt out 7.5/8 decks, spent less time shuffling, and getting more hands and more bets widespread across the entire casino floor, 24 hours per day. Then you should go to your bosses and say "You know how you're paying me to 'check' blackjack players? Yeah you could make 10x the money if you actually stopped worrying about them so much and paid me to analyze how you don't deal out enough hands before shuffling."

I had a PM with someone that is smart and someone I respect wondering just how much this would actually be. It got me wondering to do some simple napkin math to see... Follow along with me won't you? I hope/assume I'm not breaking any PM rules by quoting and sharing what I WROTE back because I think it would be good information and contribution to this thread:

When I came to WoV I literally ready every single thread on every single page in the blackjack sub-forum. I'm pretty sure I recall someone actually did this and used fair counts/results and it was in fact a fairly drastic increase for the house. Of course this is from a thread I read X years ago that was already Y years old, and it will depend on the size of the casino (i.e. number of tables) and popularity of said casino (active tables vs dead tables)...

However, if you simply take a "fair average" from what we know, we can do some simple napkin math at least...

1) ~3.2 cards per player.
2) Dealing out an extra deck and a half is another 78 cards. This would provide a little over an additional 24 hands per shoe.
3) If an average bet of $25 is used (and I think this is a little low when averaging the entire casino) that's an extra $600 per shoe in action loss.
4) An "average" to slow table gets usually 3 shoes per hour. This means each table is losing $1800 in action per hour.
5) If a joint has 5 (or only 5 'active') blackjack tables then that's an extra $9000 in action per hour.
6) If we can agree on an "average" player being about 3% negative, and I personally know of several casinos that their "average" is 3%-5% because of how bad most people play the game, and we're not even CONSIDERING most have just as many if not more 6/5 at this point, then said casino with above conditions is losing $270/hour.

7) Now let's really get detailed and figure about 12 hours per day of the above "average" conditions holding. That means said casino is losing $3240 per day for simply not dealing out an additional 1.5 decks.

8) Now let's get unnecessarily detailed and figure only 4 days per week the casino is hoppin' with the above conditions, that means said casino is losing $12,960/week for not dealing out an additional 1.5 decks.

9) Now let's just for giggles go... okay well then in 1 year said casino is losing $673,920... for simply not dealing out an additional 1.5 decks.

But yeah... I'm sure they lose almost $700k per year to card counters that are easy to spot and back off with little to no effort from a drone floor-man that makes $15/hour. Casinos are morons. News at 5.

Rome’s I like your analysis. I think it was Bill Zender who did a study on this and it’s in one of his books
Also two things many players will quit at the end of a shoe so dealing deep will get them to play a few extra hands.
One small negative thing is many casinos are no mid shoe entry and this would cause players to have to wait a little bit longer. But not that big of a deal.
Always enjoy your posts

Hey!
I said "I work" in a gambling domain not that I'm a math guy or that I'm paid to "check" blackjack players.
Everything I do I do for my own personal knowledge considering the fact that I make a living as a "player". Every statistic I do is based on historical data because I have tones of historical data from a variety of games and i can get an accurate statistics based on that and compare them with theoretical data. You would be surprised to know how many differences you can find and exploit, differences that are not necessarily occurred because of wrong implemented payouts.
My question was from personal curiosity, I'm currently "working" on something related with it.
I mentioned 6/8 decks because this is the average delt amount of cards in online casinos all over the world. There is no point to debate how many cards should be dealt to increase casinos income. They are specific reasons why a certain amout of cards are burned at the beginning of a shoe or how many cads remain undealt, depending on the game, and all off those reasons are to reduce risk, not only from players side but also from staff side. Regarding on "the more cards/hands you deal the more money you make as a casino" we should consider the fact that this is already optimised at maximum by simply reducing the decision time for each individual box, changing a dealt shoe with an already shuffled shoe (would take 15 - 20 sec) and by implementing a proper minimum and maximum bet limit for each table/game/bet type (reported to each casino's players statistics).
I do not contradict you in any way, I'm aware of all the insides of this so called "industry".

Quote: Romes

I had a PM with someone that is smart and someone I respect wondering just how much this would actually be. It got me wondering to do some simple napkin math to see... Follow along with me won't you? I hope/assume I'm not breaking any PM rules by quoting and sharing what I WROTE back because I think it would be good information and contribution to this thread:

When I came to WoV I literally ready every single thread on every single page in the blackjack sub-forum. I'm pretty sure I recall someone actually did this and used fair counts/results and it was in fact a fairly drastic increase for the house. Of course this is from a thread I read X years ago that was already Y years old, and it will depend on the size of the casino (i.e. number of tables) and popularity of said casino (active tables vs dead tables)...

However, if you simply take a "fair average" from what we know, we can do some simple napkin math at least...

1) ~3.2 cards per player.
2) Dealing out an extra deck and a half is another 78 cards. This would provide a little over an additional 24 hands per shoe.
3) If an average bet of $25 is used (and I think this is a little low when averaging the entire casino) that's an extra $600 per shoe in action loss.
4) An "average" to slow table gets usually 3 shoes per hour. This means each table is losing $1800 in action per hour.
5) If a joint has 5 (or only 5 'active') blackjack tables then that's an extra $9000 in action per hour.
6) If we can agree on an "average" player being about 3% negative, and I personally know of several casinos that their "average" is 3%-5% because of how bad most people play the game, and we're not even CONSIDERING most have just as many if not more 6/5 at this point, then said casino with above conditions is losing $270/hour.

7) Now let's really get detailed and figure about 12 hours per day of the above "average" conditions holding. That means said casino is losing $3240 per day for simply not dealing out an additional 1.5 decks.

8) Now let's get unnecessarily detailed and figure only 4 days per week the casino is hoppin' with the above conditions, that means said casino is losing $12,960/week for not dealing out an additional 1.5 decks.

9) Now let's just for giggles go... okay well then in 1 year said casino is losing $673,920... for simply not dealing out an additional 1.5 decks.

But yeah... I'm sure they lose almost $700k per year to card counters that are easy to spot and back off with little to no effort from a drone floor-man that makes $15/hour. Casinos are morons. News at 5.
link to original post

Fun post. Thanks. A few reflections:

A) Your point 4) is assuming it takes no time at all to deal out the extra 1.5 decks. Your math assumes 3 shoes per hour irrespective of pen.

B) Is there another cost to consider that a place known for deep pen attracts more counters. I’m not saying it would eat into all the extra profits of dealing another 1.5 decks. But it’s a cost to be considered in calculating the benefits of a deeper pen.

C) A continuous shuffling machine gives the casino the best of both worlds right? Never pauses to shuffle without the worry of counters. (Except for CSM counters . . . )

I believe there are a number of entertainment players who believe that a CSM alters the Most Sacred Flow of the Cards in unnatural ways, so there are likely some downsides on that path as well.

Quote: Dieter

I believe there are a number of entertainment players who believe that a CSM alters the Most Sacred Flow of the Cards in unnatural ways, so there are likely some downsides on that path as well.
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CSM machines are better for the general public as it slightly increases the return.

Quote: DRich

Quote: Dieter

I believe there are a number of entertainment players who believe that a CSM alters the Most Sacred Flow of the Cards in unnatural ways, so there are likely some downsides on that path as well.
link to original post

CSM machines are better for the general public as it slightly increases the return.
link to original post

Indeed, but too many people put their trust in superstition rather than complex math.

If a great player prefers to lose on an 8 deck 6:5 shoe game rather than a 5 deck 3:2 CSM game, let 'em have a swell time and hope they come back real soon.

(shrug)

Quote: DRich

Quote: Dieter

I believe there are a number of entertainment players who believe that a CSM alters the Most Sacred Flow of the Cards in unnatural ways, so there are likely some downsides on that path as well.
link to original post

CSM machines are better for the general public as it slightly increases the return.
link to original post

I disagree. The public is at a disadvantage with each hand so as CSM's increase the hands per hour, they increase the public disadvantage.
The only time the public is not at a disadvantage is when the cards are being shuffled. Eliminate those minutes and Joe Q Public is at a disadvantage from the time he sits down until he forks over his last chip.

Quote: DRich

Quote: Dieter

I believe there are a number of entertainment players who believe that a CSM alters the Most Sacred Flow of the Cards in unnatural ways, so there are likely some downsides on that path as well.
link to original post

CSM machines are better for the general public as it slightly increases the return.
link to original post

But they play more hands per hour on a csm so I believe that would easily negate the tiny difference in house edge.

Allowing people to lose at a quicker rate is casinospeak for customer service.