Casino Bar blackjack warning
Last Update: December 17, 2002
Following is my argument that when I played at Casino Bar on May 27, 2002 and again on December 13, 2002, my results were not consistent with a fair game of blackjack. Previously somebody approached me with what he claimed was a section of computer code he said was taken from the Casino Bar blackjack game. My interpretation of that code is that if the player has a total of 16-21 and the dealer must take a third card, if that hit card will cause the dealer to bust, then it will be rejected and the dealer will get a second chance card. This second chance card is final, whether or not it will bust the dealer. To put it another way here is the logic of the code:
This is what would be known in a real casino as dealing seconds. I do not know what the game does if the player splits and my experiment ignores split hands.
It should be emphasized that I do not know if this code is legitimate.
The goal of my experiment was to disprove that Casino Bar was playing a fair game of blackjack. To do this I designed an experiment to test the frequency the dealer busted on the third card when there was a potential to bust and the player had a total of 16-21. The course of my play this situation happened 332 times. The following table shows how many of these 332 occurrences the dealer busted on the third card, according to the dealer 2-card total.
Assuming an infinite deck for the sake of simplicity it is easy to calculate the probability the dealer will bust with any given total of 12-16. With a total of 12 there are 4 cards that will break the dealer and 9 that won't so the probability the next card will break the dealer is 4/13. Likewise the probability of busting on the next card with a total of 13 is 5/13, and so on. The next table shows the expected number of times the dealer should have busted in this experiment based on these probabilities and the number in the sample for each total from 12 to 16.
The number in the lower right corner shows the expected number of busts is 149.38. The actual number of busts was 89. This is quite a disparity. To determine the probability of this disparity I first had to calculate the variance of the number of busts to expect. Using the formula var(x+y) = var(x)+var(y)+2*cov(x,y) we can individually calculate the variance for each total. The covariance is 0 because there should be effect on one hand to the next.
Analysis of Results
The variance of the binomial distribution, which this experiment follows, is n*p*q, where p is the probability of success and q is the probability of failure. The total variance is then 84*(4/13)*(9/13) + 61*(5/13)*(8/13) + 67*(6/13)*(7/13) + 61*(7/13)*(6/13) + 59*(5/13)*(8/13) = 78.11. The standard deviation is the square root of this number, or 8.84.
The difference between actual and expected dealer busts is 149.38-89 = 60.38. This is 60.38/8.84=6.83 standard deviations below expectations. The probability of falling this far or more to the left of the bell curve is 1 in 238 billion. To put this in comparison the probability of hitting the Power Ball is 1 in 80,089,128. It would be 2976 times easier to win the power ball with one ticket than to have results this bad in a fair game.
My results have been corroborated by three other webmasters.
The GameMaster did his own independent test. Of the 223 hands in the GameMaster's sample where the player had 16-21 and the dealer had a 2-card total of 12-16 the dealer should have busted on the third card 100.77 times, but in fact only busted 53 times. The probability of 53 or less busts is 1 in 43 billion. Should there be any doubts the GameMaster videotaped his play.
Dan Pronovost, the webmaster of Deep Net Technologies , did a smaller sample of 99 hands with a player total of 16-21 and a potential dealer bust on the third card. His results show 45.54 expected busts and 28 actual busts, with a standard deviation of 4.84. The probability of observing 28 or less busts are 0.014%. I attribute this greater probability to a smaller sample size. To gather the 99 hands to meet the conditions of this experiment Dan played 500 total hands, and took a screen shot of every one. The details of his experiment can be found at deepnettech.com .
My friend M.N. also conducted various tests on the blackjack game of Casino Bar and their sister casino Casino on Air. The most convincing of which is the distribution of the dealer's third card when the dealer had a 2-card total of 12-16 and the player had 17-21 at Casino Bar and 16-21 at Casino on Air. Following are the results.
Note how heavily weighted the low cards (A-5) are compared to the high cards (9-K). Putting this distribution through a chi-squared test the chi-squared statistic is 97.83 at Casino Bar and 87.86 at Casino on Air, both with 12 degrees of freedom. The probability of a result this skewed is 1 in 676 trillion at Casino Bar and 1 in 7.8 trillion at Casino on Air.
I wish to note that I have never played at Casino on Air since they left Starnet, so I can not corroborate the Casino on Air results.
Casino Bar ResponseShortly after I posted my study I received a letter from the Casino Bar attorneys who denied my allegations, saying in part "Your report is tendentious and is of a slanderous nature. We can hardly comprehend how you could possibly reach these incorrect and misleading conclusions." In the interests of fairness I temporarily removed my report to give Casino Bar time to investigate my findings. During this waiting period, I posted at the request of the Casino Bar attorneys our exchanges.
On June 23 I received a report from the Casino Bar attorneys by Yair Tauman, PhD, Hebrew University, a leading professor of game theory in the economics department at Stony Brook State University of New York. Here is his report in its entirety.
(1) I agree with calculated probabilities of Mr. Shackleford. I also agree that experimental results reported by Mr. Shackleford are extremely unlikely under the hypothesis of a fair dealer, but at the same time his data is very unlikely to be generated under his own hypothesis as I will explain in paragraph (5) here below.
(2) I myself ran experiments on Casino Bar?s site and I derived very different results. I played 1313 hands until I obtained 400 relevant situations (where the player had total of 16-21 and the dealer had a total of 12-16). The results I arrived at are showing in the following table:
* Out of the 400 relevant situations
The table certainly indicates that the results match a fair dealer and are very unlikely to be generated by a "cheating" dealer.
I ran (with the
help of a colleague from MIT in
(4) The average return of Mr. Shackleford was 95.7% on his 1245 hands. This outcome is still statistically possible (with a sample of 10 rounds each of 1245 hands), provided that Mr. Shackleford is an experienced player and that he played his hands perfectly. However,
(5) The data provided by Mr. Shackleford is a little puzzling if we take his hypothesis about the "cheating" dealer seriously. We can use the same method of analysis he used but in a different way. If Casino Bar really were cheating as described by Mr. Shackleford, then one can calculate the chances of busting. With a total of 12 it is (4/13)?, because it would have to be that there was a bust on both of the next two cards (with the "second chance" method). We can draw a similar table to that of Mr. Shackleford
The variance of the new distribution is 84 * 0.0947 (1-0.0947) + 61 * 0.1479 (1-0.1479) + ???.= 52.26, giving a standard deviation of 7.25. Now the observed total was 89, which is (89-71.26) / 7.25 = 2.44 standard deviations higher than the expectation. This happens with a probability significantly less than one in a hundred. This shows that the data provided by Mr. Shackleford does not match his own predictions and that his hypothesis of a "second chance" method has no basis.
My own experiments show that the dealer of Casino Bar Black Jack game is fair. Out of 400 relevant situations the dealer busted on the 3rd card slightly more than he was expected to. The average return documented by Casino Bar is about 97.6%, which is very reasonable for an average player. Finally, the hypothesis of Mr. Shackleford that Casino Bar is cheating by the "second chance" method should be rejected by his own data, with a significant level of less than 1%.
First let me say I respect Mr. Tauman and his report.
I was very pleased to read Mr. Tauman's opening statements in point 1 agreeing with my calculated probabilities and that my results were "extremely unlikely under the hypothesis of a fair dealer."
In point 2 Mr. Tauman reports that he received a fair game. This I do not dispute. My allegation is that when I played on May 27, 2002, I did not get a fair game. Furthermore three other independent testers shortly after that date also evidently did not get fair games either. Casino Bar has never directly alleged that my data is incorrect and they have my log files at their disposal.
In point 3 Mr. Tauman says the expected results given the manner of dealing suggested by the code is a return of 93.8%, which is much less than the 97.6% actual return reported by Casino Bar. The 93.8% sounds reasonable to me and I do not claim that Casino Bar is dealing unfairly all the time.
In point 4 it is noted my own return was 95.7%, which I agree is possible in a fair game assuming proper basic strategy, which I do follow (sometimes with composition dependent exceptions). I would also argue this return is closer to the 93.8% assuming the dealer is dealing seconds than the 99.8% assuming a fair game under Casino Bar rules, which are quite good. In a sample of 1245 hands the actual return will vary from the expected return by as much as 6.4 percentage points 95% of the time, so my actual return proves nor disproves anything.
In point 5 Mr. Tauman is correct that my results do not mesh well with the method of dealing seconds I describe above. If I test my results against the hypothesis that the code is accurate my results are indeed 2.44 standard deviations above expectations, for a probability of 0.73% of being this high or greater. I would like to emphasize that my goal was not to prove that the code is accurate, rather to disprove a fair game.
I am willing to do one free retest of Casino Bar's blackjack game at their request. I also may do a voluntary retest even if they don't ask. My account is evidently still open, which I appreciate.
The Dispute ContinuesFollowing is a letter I received from the Casino Bar attorneys on June 28, 2002:
Dear Sirs,Following is my response sent by my attorney
Dear Mr. Levit:
On September 6, 2002, I returned to Casino Bar to see if the dealer was still dealing seconds. I think it was sporting of Casino Bar to leave my account open all this time. In 106 hands in which the player had a 16 to 21 total and the dealer had a 2-card total of 12 to 16 the following table shows how often the dealer busted on the third card.
The above table shows a total of 51 busted hands out of 106 possible. The next table shows how many are expected assuming an infinite deck.
The above table shows that in this sample the expected number of busts is 47.15. The 51 actual dealer busts are more than expected well within the range of normal variation. The probability of getting more is 22% and less 78%. So Casino Bar passed the second test for dealing seconds easily.
After my first retest showing I got a fair game my friend M.N. gave them another chance during a promotion. To my amazement her results were consistent with those of my original test. She took a sampling of the dealer's third card when the player had a total of 16-21 and the dealer had a 2-card total of 12-16. Her results very extremely skewed. The probability of a fair game resulting in a distribution as skewed or more as that shown is 1 in 6.3 trillion.
To confirm the Custom Strategy Cards retest I tested Casino Bar again on December 13, 2002. However I speculated that the system may have been programmed to give me a fair game. So I played from a friend's home on his account, which I funded. My test was the same as my first and second tests, the frequency of the dealer busting on the third card when the player had a total of 16 to 21 and the dealer was in danger of busing on the 3rd card. Following are my results.
The above table shows a total of 43 busted hands out of 160 possible. The next table shows how many are expected assuming an infinite deck.
So out of the 160 hands in the sample the expected number of busts was 73.46 and the actual number was only 43. The standard deviation of the number of busts is 6.16. My results were 4.94 standard deviations short of expectations. The probability of having 46 or fewer busts in a fair game is 1 in 2.6 million.
This time I videotaped my play, should my results ever be contested in court.