Yesterday, I posted a question about Blackjack rules. The question did not have to do with the full game which is complicated, but only with the payout rules after distribution of the first two cards. For completeness, I’m going to show the arithmetic associated with the different “edge” given to the house when the house uses an 8 deck shoe vs. a 1 deck shoe. Computation of the payout after 2 cards requires use to estimate a) the probability the player gets a blackjack but the dealer does not, b) the probability the dealer gets a blackjack, but the player does not. These happen to be equal which makes the math easier. Intermediate calculations are c) the absolute probability of a black jack on two cards and d) the conditional probability of a second jack after the player 1 gets a blackjack.
Preliminaries
The number of decks in a shoe will be indicated by “N”.
There are 52 cards in a deck. Of these 4 are aces and 16 have values equal to 10. The values of the other cards is unimportant for computing the payout after 2 cards.
For purposes of discussion, I’m going to assume the table plays at 3:2.
Probability of a blackjack
When 1 player draws 2 cards from a deck, he can get a black jack two possible ways: He can draw a 10 card followed by an ace or he can draw an ace followed by a 10. The probability of drawing a ’10’ as the first card is 16*N/52*N= 16/52. This is because there are “16*N” cards with values of “10” and “52*N” cards in the shoe.
The probability of drawing an Ace after drawing the 10 is 4N/(52*N -1). This is because there are “4N” aces in the shoe but the total number of cards in the shoe was reduced by 1.
You can repeat this reasoning to find the probability of drawing an ace followed by a 10, and sum the two to obtain the probability of a blackjack on two cards which is:
Doing this, it’s possible to show that the probability for a blackjack when drawing from a 1 deck shoe is 4.83% while the probability of drawing a blackjack from an 8 deck shoe is 4.75%.
So, you will draw a larger number of blackjacks from a 1 deck shoe.
Of course, the dealer has the exact same chance of drawing a blackjack. But, since you will win 3:2 in when you have a blackjack and only lose your wager to the dealer when he gets a black jack, all other things being equal, increased blackjacks tend to give, you, the player the edge.
However, we still don’t have enough information to compute the edge on two cards.
Conditional probability of second BlackJack
If you are playing blackjack, there is a problem: If you get a blackjack, and the dealer also gets a blackjack, your bets cancel. So, to compute the payout, we need to know how both you and the both get blackjacks.
Well it’s possible to show the probability the dealer gets a blackjack (event BJD) given that you got a blackjack (BJ) is
The values in the numerator are the number of Aces and 10s in the deck after you get your blackjack and the numbers in the denominator are the number of cards in the deck after 2 or 3 draws.
It’s possible to show that when dealing from a 1 deck shoe, the probability the dealer will draw a blackjack when you do is 3.67%; when drawing from a 8 deck shoe, the probability is 4.61%.
So, the dealer is more likely to interfere with your win when drawing from an 8 deck shoe. This makes an 8 deck show even worse for the player than we would have thought based on the reduced number of blackjacks because only does the player get fewer blackjacks, but the dealer spoils your win more often!
As a player because you will win the 3:2 payout only if you draw the black jack, and the dealer doesn’t. That probability is
So, who has the edge after 2 cards?
To compute the edge, we need to know that the player wins 3:2 when he has a blackjack but the dealer doesn’t and loses his wager when the dealer gets blackjack, but the player doesn’t. It’s easy to show the probability at which the player loses is his wager after 2 cards is exactly equal to the rate at which he wins at 3:2.
So, for each dollar wagered, the edge is:
For a 1 deck show, the player has a $0.0232 edge for each dollar bet; with an 8 deck shoe, the player’s edge drops to $0.0226.
Are you puzzled to read the player has an edge?!
Well, really they don’t. This would be the edge if the game stopped right here, and the player was allowed to take back their entire wager.
You can’t do that in blackjack.
What really happens more than 90% of the time neither the dealer nor the player gets a blackjack, and you need to start making decisions about hitting, standing, doubling, splitting etc. The house has an edge in that part of the game, and computing that edge is complicated. In fact, we can only compute it if we assign a strategy for the player.
Blackjack is a fairly popular game. It’s know that if the player makes no mistakes, and has no information about what cards remain in the deck, the edge at the end of the game is always in the favor of the dealer. Needless to say, the players lack of perfect skill further tips the edge in favor of the dealer.
It does appear that having fewer decks in the shoe gives the house a smaller edge than having a larger number of decks in the shoe. (It will also favor unskilled players if only because getting blackjack on 2 cards is pure luck.)
I can’t present a closed form solution to show that fewer decks reduces the house edge against skilled. However, in comments, Jason gives the qualitative explanation:
As I understand your inquiry, you are trying to figure out why a seemingly symmetric change to the rules (the size of the deck) has an asymmetric result (larger decks favoring the house).
The argument about blackjack [it is less likely with a larger deck, and has asymmetric payouts] is correct, but my recollection is that is represents less than 50% of the disadvantage that comes from the larger deck.
The larger asymmetry (in this case) results from the player having a choice as to when he hits or stands, while the dealer does not. In a game with fewer decks, a player knows more, and gains a greater advantage from this choice, especially in those situations where basic strategy calls for the player to stand (or double down) with less than 17 [holding a 15 against a 5].
For a concrete example of this: In a game with 1 or 2 decks, if you are holding a 9 against a dealer’s 2, you should double down. In a game with 4,6 or 8 decks, you hit. This is because the three known cards provide more information in the case of a smaller deck (increasing the chance that you will draw a ten by a greater amount and decreasing the chance that the dealer will draw to 21 by a greater amount) and this additional information is sufficient to change the optimal decision.With certain obscure exceptions, the optimal strategy does not change between six and eight decks, but the edge held by the player (mostly in those situations when you are betting that the dealer will bust) does increase.
This seems correct, and agrees with the result by those who have gone through the tedium of running the monte carlo for different player strategies.
Am I going to go out and play Blackjack?
Now that I know this blackjack trivia, I’m still not going to find an online casino to bet, bet, bet! In the first place, I don’t have any intention of memorizing all the arcane rules for blackjack to optimize my play. In the second, even if I play perfectly, I will still lose.
The only real hope would seem to be learning to count cards. That’s a lot of work, and unless you can figure out some way to play fast, and not get kicked out the casinos when you start winning, it’s not worth the time required to train. (That is, unless you like gambling.)
Still, it’s sort of interesting. But here’s a more interesting bet: Wanna bet the google ads will have a gambling flavor?
All I know is, if you are up $$$ because of a dealer bust streak, get away from the table immediately if they change dealers. Take your money and run! 😉
Andrew
What’s a dealer bust streak? They busted a whole bunch of times in a row?
Why would changing dealers matter?
Lucia,
It’s mostly just something related to a story that included me and 2 of my buddies when we were in Vegas one time. The dealer busted several times which made the table very $$$ happy. I don’t know if it was the normal dealer switch or because someone upstairs thought there was too much player winning going on, but the dealer was replaced and me and my buddy Woody decided if there was ever a time to pocket your winnings, a dealer change is an appropriate time as any. It may have signified a change in the “atmosphere”?
My buddy Rando who stayed at the table proceeded to lose his -ahem-, rear end. 😉
Andrew
Lucia: your
.
Wanna bet the google ads will have a gambling flavor?
.
Close. Real close. The Google ads I see now are for “Day Trading Stocks” and “Dating”.
Actually, it is possible to win in a casino.
.
If I go into a casino, and play the machine called “ATM”, then walk straight out, I am a winner every time.
.
Conversely, if I stay in the casino, no matter how many times I play “ATM”, all those winnings a “ATM”, seem to be canceled out at the tables.
Les,
For me, the ads above the comment line say “Top 10 Real Cash Casinos” and “MIT Blackjck Team: Learn to count cards from”.
The add across the top banner is for technical analysis.
lucia,
I humbly suggest that you are missing part of the motivation for playing blackjack.
Even if you know there is a fixed long term cost at playing optimal strategy (non counting) blackjack, the variance is very high, so you have a fairly large chance of walking away “up” after a finite amount of play.
Combine this extra bit of spice with the sheer challenge of playing a game which does take mental effort…
In short, it keeps lots of people occupied for long periods of time — at what they deem an acceptable cost for entertainment!
oliver,
“In short, it keeps lots of people occupied for long periods of time — at what they deem an acceptable cost for entertainment
This is a good thing??
kuhnkat,
Why, should they be surfing the ‘net instead? 😉
Lucia: well, I still think that “day trading” and “dating” are much more of a gamble than blackjack.
.
In blackjack, there is less chance of humiliation, too….
Kuhnkat
As a person who knits for fun, I can hardly criticize people for hobbies that “waste time”. Let me assure you that many hand knitters spend more on yarn than they would for simply dressing themselves decently. ( I could explain more. to some extent, they end up with nicer sweaters than they would ever, ever buy, But still. . . when it comes down to it, people knit $200 sweaters because they enjoy the process of knitting, not because they are saving money. It’s just another form of entertainment.)
Oliver– I actually get that part of the hobby. It’s just…. I‘d a) spend the time knitting, b) write a script to let someone test their ability to count cards, or c) surf the net, than actually gamble in a casino.
Lucia,
I was being half-facetious about the time use thing. 🙂
But I stand by the thing about variance which I said in reply to
There’s mathematically quite a decent chance that you will win if you play perfectly, and that chance is much more decent than with most any other game you can play against the House!
Lucia, there are plenty of books on the issues and code.
http://www.bjmath.com/bjcomputer/computer/gamegen.htm
http://www.bjmath.com/bjmath/feature.htm
see the 1956 article
Steve–
I know there is plenty of code/ discussion etc. I just meant I would think that for me it would be more fun to write codes to do something related to blackjack than to spend the evening at a casino playing blackjack! This doesn’t mean I wouldn’t be reinventing the wheel.
OHhhh. I like that 1956 article. It’s actually written for people who have no idea what the rules or terms are!
Steve–
Out of curiosity, I googled around. Not surprisingly, there is tons of software for every possible gambling need.
The question I most wanted to know the answer too: What can you get for free?
There seems to be software that computes odds with different strategies (as you already found.) More importantly to anyone who eventually might want to step foot in a casino, software to let you ‘play’ to make sure you can actually manage to apply your strategy when the cards are flying furiously. (After all, there is a difference between theoretically knowing the strategy and being able to implement it quickly. )
Plus, in between, there is software that lets you play and simultaneously tells you what the strategy is dictating for your play.
Hmm… I am seeing gambling ads from google. Not too surprising of course.
I like the internet poker sites where they just give you $10,000 for nothing and you can play for free.
edward — Those are fun, but people tend to play ridiculously ‘loose’ there since there’s no (financial) pressure to play with any caution.
Lucia,
This is only slightly related but you might also like
The Computer Poker Research Group at U. of Alberta, home to Polaris (a strong Limit Poker AI).
I especially like the title of the Zinkevich et al. (2007) paper: Regret Minimization in Games with Incomplete Information
I just play for the “free” drinks.
Lucia I thought I gave you a link to some free software. If you tell me exactly what you want to do I can research it for you and find the code, or find something close that you can modify. Somewhere I know you can find the tables for the Pwin/Ploss/Pdraw for single deck given a two person game where you know what you hold and see the dealer card up. I had a book with these probability tables.
Steve– I’m not planning to do anything! I was just giving “for instances” of what type of activities seem more fun or less fun relatively.
On the “no fun at all” (for me) we could put: Watching ballet, watching monster truck races, going to a casino and actually playing blackjack and having a root canal.
In contrast, thinking up “what if” questions about black jack and finding the answer by writing a script would be more fun than playing blackjack.
Also, the only way I could even imagine playing blackjack could be fun is if I devoted hoards of time to learning a strategy, counting cards etc. with the intention of undertaking the challenge to beat the house. Otherwise, the whole exercise of playing knowing the math says I lose on average would be irritating for me!
But I’m not seriously thinking of devoting hoards of work to learning how to play blackjack. Doing so would make playing blackjack less irritating than playing it incompetently… but…. I’d still rather stay home and knit!
On the substantive questions about blackjack, I think I’ve found the answers I was interested in. The information about what happens on 2 cards was not enough to state that a 1 deck shoe is better for players. Nevertheless, a 1 deck show is better for players and can be shown by people doing all the calculations for the rest of the game.
Here’s a tip I received by email. You can get rich.
you know in roulete you can bet on blacks or reds. If you bet $1 on black and it goes black you win $1 but
if it goes red you loose your $1.
So I found a way you win everytime:
bet $1 on black if it goes black you win $1
now again bet $1 on black, if it goes red bet $3 on black, if it goes red again bet $8 on black,
if red again bet $20 on black, red again bet $52 on black (always multiple you previous lost bet around 2.5)
if now is black you win $52 so you have $104 and you bet:
$1 + $3 + $8 + $20 + $52 = $84 So you just won $20 🙂
——
See it’s easy. So since alcohol fueled monster trucks don’t impress you maybe you would have fun with this.
Hmmm… I guess a) the math is simpler if we forget about the zero and b) it’s inconceivable that it will be red again?
Two quick comments:
– Jeff’s strategy doesn’t work because there is always a house limit to the maximum bet which prevents the player from doubling ad infinitum. In finance terms, Jeff’s strategy has a positive probability of winning but a negative expected value (one of my finance professors actually used Jeff’s strategy as an example to explain the difference in concepts).
– On point not being discussed is the cut (how many cards are left behind in the shoe). If the same number of cards are left behind, then the experienced player will prefer a multi-deck as there is relatively less information hidden. Experienced players generally get their edge when the multi-deck is close to the end and there has been a noticeable run in either high or low number cards.
brid (Comment#23294) November 11th, 2009 at 11:23 am
That’s right, but as a non-gambler it took me a bit to figure out after receiving the email. It’s a cute problem – better than monster trucks.
I do love the edit feature:
Not that there’s anything wrong with gambling, it just feels like voluntary taxes.
jeff id–
Exactly. The thought involved in gambling is more fun than the gambling. 🙂
I respectfully have to disagree with both of you. Vegas, Baby!
brid:
I think it takes a little more defining of terms to make the claim that a betting system (such as Jeff’s martingale) has a positive chance of “winning” yet also has a negative expected value.
Oliver,
A positive chance of winning means a that there is more than a 50% probability of you leaving the casino with more money than you started. A negative expected value means that the risk neutral expected gain (in dollar terms) is negative (calculated as the sum of the probabilities of each expected outcome multiplied by the outcome).
As a simple example, we enter into a bet whereby you roll a fair die. If the die comes up any number from 1 -5, I pay you a dollar. But if a six comes up, you pay me $100. Your probability of winning is 5/6 = 83%. But your expected value = 5/6*$1 – 1/6 *$100 = -$15.8. This is a very simple example, but the concept extends to many fairly complex financial uses such as pricing of derivatives.
brid,
I guess I didn’t make myself very clear. My point is that you do not have the edge vs. the house in blackjack; the martingale is supposed to make you a winner despite that inconvenient fact.
Once you allow for multiple die rolls “ad infinitum”, then a further decision point has to be defined for “winning”: e.g., some fixed amount of winnings; similarly for “losing.” And then you get to see where the model breaks… 😉
Oliver,
I am a little confused by your comment. Jeff’s strategy was to do with roulette, not blackjack. In any event, having a positive probability of winning is not directly related to the house edge. A casino essentially models its business as an experiment with an infinite number of trials so it is the expected value that is of importance. As a result, there is no need for an interim decision point. In a no bet limit scenario, this assumption breaks down but it works pretty well in the real world. Outside casino land, we have to build in loss given default models and risk aversion among other refinements.
If you are looking from the point of view of the player, an interim decision point doesn’t change the analysis. There is just no way the player can get a positive expected value from roulette in a game that contains a zero or double zero. Changing the definitions cannot change this. All roulette is is a series of independent, uncorrelated trials. Each trial has a negative expected value. Therefore, a roulette strategy is simply adding a series of negative expected values. This can never be a positive number. Blackjack is of course different because the hands are not uncorrelated.
brid,
My mistake, it was roulette in Jeff’s example. However, the betting strategy ought to work basically the same way whether it’s blackjack or roulette. The key idea of the martingale is that every term in the geometric sequence (1/2)^n is 1 larger than the sums of all previous terms. Hence, if you allow the game to go on long enough (and the bet sizes to grow without limit), then the gambler can always come out with a 1 unit profit if you merely double bets at every round. Hence, 100% winning percentage and expected value exactly 1 unit.
Once you define a betting limit (which is the same as setting a maximum debt or the gambler) then you will have to compute the likelihood of n consecutive losses, etc., which will stop you cold (and at a 2^n loss). Then the “house edge” does matter in the sense that the chance of actually landing red (or black) is slightly worse than 1/2 (despite the 1:1 payout structure).
Alternatively you can set a positive “win” amount goal and adjust the geometric progression accordingly; again the infinite string of bets model predicts a perfect success rate, but the model breaks down because you can’t actually make those bets.
Am I missing something here (sadly, the possibility is finite and ~1). 🙂
My friend thought he had a similar winning system to Jeff’s, except his involved the columns.
He figured that 1 column could never dominate for more than a few (say 5) goes, and the odd one would go to 10 (max). The odds after this become enormous, but one must recall that averages only work over thousands of goes, not a few.
So he tried out his system by waiting for 1 column to come up at least 3-5 times, then bet on the other 2 (making this more expensive than Jeff’s red / black, but it has a higher dividend). The dealer saw pretty quickly what he was doing and we had some good chats with him. He did say he had seen 1 column come up a lot in a row, but my friend didn’t really take much notice.
Long story short, the 1 column came up 13 times when my firend’s money ran out. We stayed there to see how long it would last, but we finally gave up when it hit 20 occurrences in a row.
That was the end of my friend trying to beat the odds. ps his rich grandma felt bad that he lost some $ and reimbursed him in full!
pete m (Comment#23308) November 11th, 2009 at 8:51 pm
My friend thought he had a similar winning system to Jeff’s, except his involved the columns.
I also know people who are convinced by this method – esp. when they win the first few times…then realise how expensive it gets when that first column gets growing :o)
At the end of the day I always have the same message for anyone who comes up with a “system” to break the casino: Look at their house, now compare it to yours.
Cheers
Mark
Oliver,
I think we are in the same place. As long there is a limit (of any size) on the maximum bet, the math will always show that there is a negative expected value to the strategy.
When you are dealing with infinity, strange thing happen. This reminds me of one of my favorite betting brain teasers. I give you two sealed envelopes that contain unknown amounts of money. You choose one at random. I tell you that one envelope has exactly double the amount of money as the other. I give you the option to switch envelopes. Do you switch? if you do, I then give you the option to switch back. Do you take this option? (warning – this puzzle is deceptively difficult and can drive you crazy).