 Ask the Wizard: |
Online Blackjack |
At BetJamaica I played 30 hands and lost 21.5 units. What is the probability of that? — Timothy Pelletreau from Medford, NJ
The standard deviation per hand in blackjack is 1.15 under Vegas Strip rules ( source). This can vary, depending on the rules, but since you didn't state them, we'll go with 1.15. So, the standard deviation of 30 hands would be sqrt(30) × 1.15 = 6.30. I don’t know what their blackjack rules are, but let’s assume a house edge of 0.4%. So in 30 bets, you would expect to lose 30 × 0.004 = 0.12 units. Your losses exceeded expectations by 21.5-0.12 = 21.38 units. That is 21.38/6.3 = 3.39 standard deviations south of expectations. The probability of that is 0.000349, or 1 in 2862. I’m afraid this doesn’t rise to the level to make any kind of accusations. If you still suspect something fishy, I would gather a larger sample size. July 11, 2008
Why does BetFair have zero house edge blackjack? I suspect that it is because they realize that players either don't know the optimal strategy or sometimes can't act on it (for example if they bet their last chip and then are unable to act on a double down or splitting opportunity). – Nick from London
They also have zero house edge baccarat and roulette, so that can’t be entirely the reason. My theory is that it is a way to get players through the door. Their main casino has many more games and higher bet limits. I'm sure some of the Zero Lounge players wander into the regular casino eventually.
November 23, 2007
Could you please make a chart for Party Poker single deck blackjack. Also, with best play, what are the odds. Here are the rules - Mark H. from New York
As I read them, the rules are:
-
1 deck
-
Dealer hits soft 17
-
Double only on 9 to 11
-
Double after split allowed (subject to 9 to 11 rule)
-
No surrender
-
Blackjack pays 3 to 2
-
Dealer always peeks for blackjack
Here is the basic strategy.
The house edge under these rules is 0.16%.
October 4, 2006
I am a part time blackjack player with a lot of success in land based casinos. I am thinking to start playing online but I have a few questions about this. Does a payout percentage of for example 98% mean that you lose 2% anyway regarding good or bad play. In European blackjack with no hole card, if you play last box isn't it better to leave the little card for the bank or must I hit anyway? Sometimes i have my doubts about this. P.S. Love your site! THANKS – Andrew from Belgium
Thanks. Payout percentages such as this are historical. For example King Neptune’s casino posts their June 2006 report at www.kingneptunescasino.com/report567.php?bTag=knaff2287. The 96.78% for table games means that in June 2006 the ratio of money returned to money bet was 96.78%. In other words an actual house edge of 3.22%. Your own results will depend on the game rules, your skill (in games of decision making), and luck. In most games the odds are quantifiable so payout reports are not useful. It shouldn’t matter to you how badly other players have played or the mix of games they chose. Where these reports are very useful is in evaluating the slots. No casino that I know of volunteers how loose their slots are theoretically set, but such payout reports gives the user a good idea. If looking at other months you see that King Neptune’s pays about 96% in slots. I also think it is a good sign of a good operation to have return percentages independently verified. It shows the casino has nothing to hide.
September 13, 2006
I was playing baccarat online at USACasino which uses a live dealer and Playtech software. Apparently Playtech has instituted a new rule that the dealer burns a card after each hand is dealt. This is not the way it is played in casinos. What bearing, if any, does this have on the odds of the game. I can't believe the casino would institute a new rule that wasn't in their favor. -- Phil from Yonkers
Burning cards has no effect on the basic strategy player. They probably are doing this to discourage card counters. However, they may as well just shuffle earlier. For card counting purposes what is important is the number of cards seen, it doesn't matter whether the unseen cards are burned or behind the cut card.
May 18, 2006
Dear Wizard, I have read your comments about online blackjack and the fact that the decks are shuffled after each hand. Is there a casino online that has less shuffling or if not, am I forced to play live dealer blackjack which has more decks stacked and at least I can be aware of when shuffling does occur. Cheers. -- Ken from Oxford, UK
Perhaps you misunderstood the point I was trying to make in my blackjack appendix 10. The bottom line is that unless you are a card counter shuffling after every hand, as opposed to using a cut card, is a good thing. The house edge is reduced 0.02% in a 6-deck game, to 0.11% in a 1-deck game, by shuffling after every hand.
March 13, 2006Since most online casinos deal each hand out of a
"fresh shoe," is there an opportunity to create a special
basic strategy card for the "first hand out of a shoe," or
do your various basic strategy cards work as well? - Bill
W from summit, New Jersey
The existing basic charts, including mine, are
already based on a fresh shuffle. Sept.
11, 2005
I was playing blackjack online. I bet all of my money
on a single hand, and was dealt a pair of eights versus the
dealer's three. Since I could not split, I exited the game,
went to the cashier, and bought more chips. However, when I
returned to the hand, my newly purchased chips were not at
the table. I spoke with customer support, and was told that
once a hand is in progress, additional chips may not be
brought to the table. However, there is nothing on this
casino's site stating that rule. -- Patrick from Jersey
City, New Jersey
If I had written the software I would have
allowed this, as it is allowed in the land casinos.
However remember that in most casinos Internet gambling
is unregulated and the player has essentially no rights.
The terms and conditions always seem to say that the
casino is always right in any dispute. In cases where a
rule is not addressed make no assumptions that it will go
your way. Sept.
4, 2005
What is the best betting system for a player to use
when playing the online casino blackjack games that give the
PLAYER the slight edge? Are there any systems when there is
a positive player edge that can be used to most effectively
gain maximum winnings with minimum chance of losses over the
long run? Or would the best system be to bet the same amount
consistently and follow the optimum basic strategy (and its
exceptions for single deck) for all the hands
played?
I would recommend flat betting. The expected
return is the same regardless of how you bet, but flat
betting is best for minimizing volatility and ensuring
bankroll preservation. Jan. 9,
2005
Are there any basic strategy exceptions for doubling
and splitting on the Internet when cash back applies to the
total amount bet (thus giving the player an extra incentive
to double or split). Please assume a cash back rate of
0.1%.
I could not find any hand where an extra 0.1%
would cause a change in strategy. The closest case I
could find, based on six decks and the dealer standing on
soft 17, is A,4 against a dealer 4. In that hand doubling
costs the player just under 0.3% in expected value
compared to hitting. So if you can get 0.3% or more then
double that hand. Other borderline hands I examined cost
the player at least 1% to double or split, and I have
never seen cash back that high in blackjack.
July 4, 2004
I read somewhere that online casinos may shuffle the
deck after every turn and that card counting then would be
of no value. How can I find out for sure if a casino is
doing this?
You could test for how often the same card
appears in two consecutive hands. Considering only the
four initial cards dealt and assuming all four are
different the expected number of those cards seen in the
next hand is 16/52 = 0.307692, assuming the cards are
shuffled after every hand. If you see fewer repeats than
would argue that the cards are not shuffled after every
hand. If the cards are not shuffled between hands the
expected number of repeats seen would be
4*(4d-1)/(52d-4), where d is the number of decks. It is
easier, but less reliable, to just ask customer support.
April 22, 2004
Is there software that will count cards as you input
them? Can this be used to play online blackjack? Or does
online blackjack shuffle after each hand?
Yes. Have a look at beejack.
Most, but not all, casinos shuffle after every hand. I
don't know if it practical to actually use this product.
If you try it please let me know how you do, I might like
to try it myself. Nov. 19,
2003
I just wanted to make sure that the correct play in
the Microgaming single deck game of A,A vs A is to split as
shown on your basic
strategy card. I'm pretty sure the correct play
would be to hit, at least with 4 decks, and it just seems
that with all the 10's left you might lose twice your bet
pretty often. Not that I don't believe you but I just wanted
to make verify. Any chance you have the EV of hitting vs
splitting on this hand? - William K. from Pittsburgh, USA
I've been questioned about this several times
and continue to maintain that despite losing the total
bet the option to draw to split aces overcomes the
European no-peek rule, thus splitting is the better play.
Based on one deck the expected value of each hand
(considering the possibility of a dealer blackjack) is
-0.532849 for hitting and -0.223277 for splitting. So
splitting is better by about 31% of a unit. Splitting is
also better for the 4-deck game, which no Microgaming
player should be playing since a 1-deck game with the
same rules is available. Nov. 3,
2002
Microgaming has a new blackjack game called "Lucky
Sevens" where you get bonuses for having your first card a
7, first two cards 7's, and first 3 cards 7's. It also has a
progressive pot which you win if you get all 3 7 of
diamonds. The bad part is that you have to ante $1 for each
hand. You have no choice on this if you play the game. My
question is: at what size progressive jackpot does this
become a positive expectation game. Thanks! - Bryan M. from
Austin, Texas
Good question. "Triple Sevens Blackjack" is
played with five decks with a $1 mandatory side bet.
Following is the side bet return table, based on a meter
of $41,227 (the amount on August 31, 2002).
|
Microgaming Triple Sevens Side Bet
|
|
Hand
|
Permutations
|
Probability
|
Pays
|
Return
|
|
1 seven
|
1238400
|
0.07128
|
5 for 1
|
0.3564
|
|
2 unsuited 7's
|
72000
|
0.004144
|
25 for 1
|
0.103605
|
|
2 suited 7's
|
19200
|
0.001105
|
50 for 1
|
0.055256
|
|
3 unsuited 7's
|
6600
|
0.00038
|
250 for 1
|
0.094971
|
|
3 suited 7's
|
180
|
0.00001
|
1000 for 1
|
0.01036
|
|
3 diamond 7's
|
60
|
0.000003
|
41227 for 1
|
0.142377
|
|
Non-paying hand
|
16037280
|
0.923077
|
0
|
0
|
|
Total
|
17373720
|
1
|
|
0.76297
|
If the player splits two sevens the connectivity is
lost and the player begins each hand with one seven for
purposes of the side bet. The player should never split
sevens in this game at low bet levels ($40 or less). The
house edge on the side bet is 37.94%, less 3.45% for
every $10,000 in the meter. When I checked at the Golden
Tiger casino on August 31, 2002, the meter was at
$41,227, for a return of 76.30%, or a house edge of
23.70%. For the side bet to have no house edge the meter
would need to reach $109,862. This does not consider the
0.0057% cost of deviating from basic strategy by not
splitting two sevens. Assuming the player is betting $2
on the blackjack hand (the minimum) the meter would need
to reach $113,196 for the combined game to have no house
edge. Aug. 31,
2002
Do you have or can you make up a basic strategy chart
for Lasseters's blackjack? I couldn't find one on the
internet. - Boyd S. from Canton, USA
Here you go...
Aug. 11,
2002
I am playing blackjack with unified gaming software.
They claim they do not shuffle after every hand especially
in the six deck, yet I have never seen it stop to shuffle.
When I called the company to ask they said it is indicated
when shuffling but I have never seen anything written on the
screen that it is shuffling. After playing for usually 45
minutes to an hour it does freeze up and one can not even
use the exit..everything shuts down. ? - Janice D. from
Muskego, Wisconsin
It is my understanding they shuffle after every
hand in the single deck game. When I have played the
six-deck game you can hear a shuffling sound and the
bottom of the screen says "shuffling" from time to time.
Perhaps you were only playing the single deck game. It
seems like they do this at random times. The lost
connection is a known annoyance with Unified Gaming
software. I mention it myself in my review. Some people
say if you disable your sound card it helps with this
problem. April 15,
2002
You stated that Unified gaming used to have a player
advantage a few months ago but no more. How did they change
their software? - Janice D. from Muskego, Wisconsin
They used to allow early surrender against an
ace. April 15, 2002
I was just wondering if Las Vegas Video Blackjack
reshuffles after every hand or after all the decks are
played. I know the tables in Las Vegas do it after all the
decks are played because if they did it after every hand no
one would play. Are the odds that you'll win worse if there
is a shuffle after every hand? Is this even legal or
necessary? - Stephanie from Evans, USA
I don't know when they shuffle but I would
speculate after every hand. From my blackjack
appendix 10 you will learn that the player's odds
improve slightly if the dealer plays exactly n hands
between shuffles (including one) rather than playing to
cut card, finishing the hand, and then shuffling.
Mar. 6, 2002
The Question I have is about Microgaming Blackjack. I
have played the for fun version of about 10 online casino
trying to determine the best one and I run into a question I
have yet been able to understand. Out of about 1000 games or
more at each casino I found out some days using the basic
strategy for the casino from your web site I stay just about
even with the dealer and some times show a small credit
profit. But other days I am lucky to win 20 hands out of
100. And I never stray from the strategy and just bet an
even bet. Is this normal that some days the random generator
is set up to sway the dealers way for a long period of time
and if so does it ever sway the players way that much too? I
sure would like to know more because I really enjoy the game
and am not wanting to make a lot of money but when you are
betting min. and it eats away 200 credits in no time this
does not seem normal odds. Please if you can educate me on
this I would think you very much. - Dean V. from Winston
Salem, USA
In my strong opinion the variation you're
observing is the result of random variation in the cards,
not where you're playing. I suspect you are exaggerating
about sometimes only winning 20 hands out of 100, that
would be very unlikely. Some people believe that
Microgaming has a "take down" mode in which the player
will lose like crazy for a period of time. It is to be
expected that in any game the player will occasionally
have bad losing streaks, as well as good winning streaks.
So these alleged "take down" periods I think are just
normal bad luck and that all Microgaming casinos deal a
fair game. Feb. 11,
2002
I'm pretty sure this isn't possible, but is there any
way to count cards when playing blackjack in an online
casino? P.S. Your site is great. Being a beginner it has
given me lots of good advice. - Dustin from Austin,
USA
Thanks for the compliments. Most online casinos
shuffle the cards after every hand. At single deck
casinos (Boss Media version 1, Microgaming, Unified
Gaming) you can use the cards already on the table to
alter your play sometimes. See my blackjack
appendix3A for all the details. There are some
casinos that play into a shoe a little way but not far,
and have restrictive betting limits. I have yet to be
convinced of any worthwhile opportunity on the Internet
to count cards. Nov. 23,
2001
First of all, thanks for your very informative,
comprehensive, and overall helpful site. I am wondering
which, if any, online casinos currently advise the player of
a shuffle in blackjack (multi-deck, of course). Also, do you
know, among the majority who do not, which shuffle after
each hand and which just do not advise of a shuffle
(although it actually occurs after many hands)? It would be
great to have this knowledge. A follow up question would be,
if they do indeed shuffle at regular casino intervals, can a
player assume that if he enters a private table that he
beings with a full shoe? Thanks again for your great web
site, and I look forward to your response to my questions. -
Tony M. from Columbus, Ohio
[answer missing].
April 15, 2001
Hey Wizard...I have just got into online gambling and
have referred to your site for pretty much everything...I
appreciate all the information on the single deck Unified
Gaming information, but what about the six deck? Can you
tell me the player (or dealer) edge in this game as well as
a basic strategy? It would seem to me that this would be one
of the best bets at which to count cards...what do you
think? Thank you in advance! - Chris P. from Denton,
TX
You should follow my multiple-deck strategy but
surrender against aces in the same hands as the
single-deck strategy. I havenít studied if
counting this game is more profitable than the
single-deck game but I would be interested to hear from
anyone who is doing as you suggest.
Feb. 10, 2001
I notice that some internet Casinos give you 2 cards
right away one on each hand before you begin to play each
hand, unlike a live casino which play the split, one hand at
a time. Is this a house advantage dealing two cards before
beginning play on the split hands? - Karter from Calgary,
Canada
Mathematically speaking it doesnít make
any difference. Jan. 20,
2001
I want to know the best blackjack strategy card for
Gamblingsoftware. The Wizards of Odds page doesn't carry a
table for this software, but the Java-based games here are
becoming increasingly popular. ñ Nevin from the
Philippines
You can correctly use the Cryptologic
basic strategy to play at a GamblingSoftware.com
casino. Oct. 5, 2000
When playing online blackjack, how do you tell when
the deck is shuffled? I play Microgaming casinos (which you
report as using 1 deck), but I do not know if each time I
play if it is a new deck, and there are no signs of knowing
when the deck is shuffled. ñ Brian of State College,
USA
Most online casinos shuffle after every hand.
Others shuffle at random times but do not indicate
exactly when to the player. I have noticed Microgaming
casinos flash the word "shuffling" about one hand in
four. However if you track the cards between these
announcements you will sometimes see the same card twice,
which is impossible in a single deck game, assuming you
believe them about when they shuffle. As far as I know
they actually shuffle after every hand but for reasons I
do not understand only indicate a shuffle occasionally.
If I remember correctly Cryptologic casinos do indeed
indicate when they are shuffling their eight deck
shoe. Nov. 19, 2000
Just one more question today. In your introduction to
your site, you make a big deal of losing in the long run at
all games of chance. However, what is your opinion of
playing tons of hands at Unified Gaming sites with the -0.57
house advantage? How realistic would it be to make $5.70 per
hundred hands with $10 bets for a few hours a day? It would
sure beat getting another part-time job:), even if I could
only do it for a few months (while I'm finishing school at
the U.of U.). The math seems correct even to me (a history
major), yet I have this nagging feeling that I've left
something unaccounted for, and this wishful thinking is just
too good to be true. I'd appreciate your opinion. ñ
Dave P. of Provo, USA
Your math is correct. For every 100 bets at $10
you make you could expect to make $5.70 assuming you
follow my basic strategy. Keep in mind this is a
long-term return and in the short run you could easily
lose. There is also a problem with broken connections
with Unified Gaming casinos, which slows down the number
of hands playable per hour. Somebody told me that turning
off the sound card can take care of this problem.
Whatever your gambling bankroll is I would recommend
dividing that by 100 and bet in those units. So to bet
$10 a hand you should have $1000 to risk.
Nov. 4, 2000
I have never gambled on-line, but I would suppose that
since you play the games from your computer, with no one
watching, it would be a lot easier to do things as card
counting or even using a computer to make your decisions. Is
this true? Do on-line casinos have any way to check if
you're cheating? Also, do you have a time limit to play each
hand? - Gonzalo from Mexico City, Mexico
Yes, you can easily count cards when playing
blackjack online. In fact some products like Blackjack
2021 are specifically designed to aid the player
doing this. It isn't cheating either. However most online
casinos shuffle after every hand which makes counting
cards a waste of time. I have heard that a few have
penetration of about 1/3 of the shoe but can't name any.
Most places seem to have no time
limit. Oct. 5, 2000
Q: I want to play blackjack at Lasseters Casino. They
use Access software i.e. Infinite number of decks ,split any
pair and resplit once, double on any two cards, double after
split and European hole card rule, and dealer stands on soft
17. Please supply me with the relevant Basic Strategy chart.
Thank you. - Peter H. from Hemel Hempstead, United
Kingdom
A: Here is my Lasseter's basic
strategy. Sept. 3,
2000
Q: I have just started gambling online and I am having
a difficult time believing that these online casinos are
really fair. although I have tried just three different
casinos, they seem to be very hard to beat. I am a fairly
consistent winner in Vegas at Blackjack, but these online
guys are tough. So the question is-- are the online
Blackjack games really fair. Thanks and by the way, I really
enjoy your site. I have learned a great deal and am looking
forward to applying some of your knowlege during my next
trip to Vegas. thank again!! - Larry R. from Nucla, USA
A: Thanks for your kind words. I think that the
vast majority on online casinos are fair. However I won't
claim that all of them are. Check out my casino
blacklist for casinos I had problems with. Assuming
you aren't playing one of the few bad ones I would
suggest your losing is just the result of bad
luck. Aug. 20, 2000
Q: What are some of the best online casinos to play
blackjack on, with the following criteria in mind - Best
player odds, fewest decks, fairness? - Robert from Hamilton,
Canada.
A: Assuming you put the greatest priority on
player odds I would have to go with a Unified Gaming
casino. You'll enjoy a player edge of 0.57% assuming you
following the proper
strategy. I list of such casinos can be found
here.
Aug. 13, 2000
Q: Are you confident of your figures on the online
software? In particular, do you really have a .53% advantage
over the house just with good basic strategy? I've never
seen anyone give figures of a player advantage before for
any rules with only basic strategy. - Jack Z. of Olympia,
Washington
A: I'm confident with that figure. It was
determined using a random simulation program of mine. I
speculate the creators of the software made a mistake and
only intended to offer late surrender on tens and aces,
as was the case with the preceding version. I called one
online casino that uses Unified Gaming and they were not
even aware the new software allowed for early surrender
on aces. I give the house edge figures as a service to my
visitors. July 16,
2000
Q: Using your basic strategy for Cryptologic, what is
the probability of winning 5 units before losing 20. If I
played everyday for a year getting out after winning 5 units
or losing 20 units, what can I expect my net to be? - Bert
W. of Richmond, Virginia
A: There was a similar question asked in the
June 15 column, which I worked out in detail. However
briefly I estimate the answer to be about 78%. Over a
year you could expect to lose 0.52% (the house edge) of
the total initial bets. July 2,
2000
Q: What is the best site to play free online
blackjack? - Heather from Alma
I never play for free but Boss Media casinos
have a very nice game you can play for free. To name just
one try the Riverboat
Casino. July 2,
2000
Q: I was wondering... Knowing that online casinos
shuffle after every hand, making card counting impossible, I
have found that there are, indeed, a few advantages. A) You
do not have to tip, B) You can go one-on-one against the
dealer (no other players), C) You can play more hands
(again, due to the lack of other players) her hour than in a
real casino, and D) you can enjoy a large betting spread
without suspicion. Do these factors make the online game
advantageous in any way? Has any test been run on these
systems? So far, I've done nicely -- in fact, I've done
better online than in any real casino. What do you think?
Any input is appreciated. Thanks! - Nathan S.
You have already pointed out some of the
advantages of online blackjack as opposed to a physical
casino. Unless you are card counting it is to your
advantage that they shuffle after every hand. In the
casinos they will play until a specific point is reached
in the deck, finish that hand, and then shuffle. This
works to the player's disadvantage because more cards
will be dealt when the shuffle point is reached in a deck
rich in small cards than one rich in big cards. In other
words in a physical casinos the player will see slightly
more small cards than large cards over the long run,
which is bad for the player. However much more important
is the fact that you don't have to feel obligated to tip
in an online casino. The faster play, one on one play,
and the larger spreads do not help nor hurt the casual
player. I attribute your better results in online casinos
to any combination of luck, better rules, and not having
to tip. April 29,
2000
Q: Just set up an account with Islandcasino today. On
their Perplexa BJ games, what are the shuffle points for the
one- and six-deck games? I presume one deck is shuffled
after ever hand. What about six decks? - Jay of USA
A: Both games are both shuffled after every
hand. Thanks to Darla with Gambler's Palace (another
Perplexa casino) for her help on this one.
Mar. 18, 2000
Q: What is the minimum blackjack bet at most online
casinos? Are there a time limits in which to make your
decisions at the blackjack tables? If so, what are they? -
Reg A. from Vancouver, British Columbia
A: The minimum varies from place to place. At
Perplexa casinos it is $1, Microgaming is $2, Starnet and
Cryptologic are $5. If you are playing by yourself then
you can take a long time. I've had phone calls in middle
of a hand, came back 15 minutes later and resumed the
hand without having been logged out. You will get logged
out for inactivity eventually but it takes quite a while.
However if you are playing at a group table there is a
time limit per decision of about 30 seconds. At the Sands
of the Caribbean you can see your clock ticking down.
March 11, 2000
Q: Do you really trust the online gambling sites for
such games as blackjack. I played one a few weeks back. The
dealer got 14 blackjacks to my 3 during that session. I'm
just suspicious and interested in your comments. Thanks. -
Mitch
A: What you should be worried about from online
casinos is not that they are cheating with the cards. The
casinos don't need to cheat to make money, the built in
house edge is sufficient to ensure financial security. I
speculate that if they did cheat they would actually make
less money because their players would not return.
However I do not claim that nobody is cheating, anything
is possible. What you should be worried about is bad
customer service. Some online casinos have a lot to learn
about answering e-mail timely and understanding their own
rules. Regarding your question about getting only 3
blackjacks out of 17 the probability of getting 3 or less
is:
(1/2)17*(combin(17,0)+combin(17,1)+combin(17,2)+combin(17,3))
=
(1/131072) * (1 + 17 + 136 + 680) = 834/131072 =
.0063629 .
In other words this is going to happen to 1 person in
157 on average. Still this is nowhere near the kind of
evidence needed to fairly acuse foul play. With any kind
of gambling your money will go up and down like a roller
coaster. To truly see if the long run looks fishy you
need to log thousands of trials and their outcome. I may
be willing to analyze the results of such data if it were
summarized properly. March 4,
2000
Q: According to your bj software tables, both the
perplexa and boss media systems give the player a slight
edge. If this is so, how can the casinos that use these
software systems turn a profit? Is it because relatively few
players use basic strategy and/or good money management
techniques? -trekon3 from ?
A: Online casinos are not the only ones to offer
positive expectation games. Using basic strategy some Las
Vegas casinos offer games with a negative house edge.
There are video poker games all over the country with a
positive expectation assuming optimal strategy. The
reason the casinos can afford to do this is that the vast
majority of players make errors in strategy. As someone
who has played hundreds of hours of blackjack in casinos
all over the US I seldom see other players playing proper
basic strategy. Money management has nothing to do with
it. March 4, 2000
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