Last Updated: December 17, 2002

# Casino Bar Blackjack Warning

## Introduction

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:

1. If player has total of 16-21 go to step 2, other wise play normally.
2. If dealer's 2-card total is 12-16 then go to step 3, otherwise play normally.
3. Peek at the next card in the deck, if it would bust dealer then burn it and take following card, otherwise give it to dealer.
4. Take further cards as necessary to attain total of 17 or more, then score hand.

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 Experiment

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.

### Casino Bar Experiment Results

Dealer 2-card Total Bust on Third Card
Yes No Total
12 11 73 84
13 13 48 61
14 18 49 67
15 21 40 61
16 26 33 59
total 89 243 332

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.

### Casino Bar Experiment Results

Dealer 2-card Total Sample Total Probability of Bust Expected Busts
12 84 30.77% 25.85
13 61 38.46% 23.46
14 67 46.15% 30.92
15 61 53.85% 32.85
16 59 61.54% 36.31
total 332 149.38

## Analysis of Results

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.

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.

## Independent Tests

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.

### Casino Bar Third Card Distribution

Card Casino Bar Casino on Air
A 48 52
2 54 45
3 71 45
4 42 58
5 46 45
6 39 43
7 33 32
8 36 37
9 24 24
10 20 12
J 13 19
Q 20 12
K 15 16
Total 461 440
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 Response

Shortly 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.

June 23, 2002

(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:

 Dealer?s card total Total* Busts on 3rdcard* Theoretical expected # of busts Busts on 3rdhigher card 12 82 31 25.23 43 13 82 26 31.54 38 14 81 36 37.38 44 15 83 48 44.7 54 16 72 45 44.3 45 Total 400 186 183.15 224

* 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.

(3)     I ran (with the help of a colleague from MIT in Boston) a simulation program of playing Black Jack with a "cheating" dealer and with a fair dealer. We ran 10 samples of 1245 games each (the same number of games that Mr. Shackleford played on Casino Bar). We assured perfect conditions for the player, such as splitting similar cards was allowed more than once (this is not the case with Casino Bar). The expected return of a perfect player with a fair dealer was about 99.2% while with a cheating dealer is was about 93.8%. The average return of a player in Casino Bar is 97.6% which is a very reasonable outcome for an average player (not a perfect player) under a fair dealer and extremely unlikely for a "cheating" dealer.

(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

 Dealer?s card Total Total obtained by Mr. Shackleford Probabilities under the 2ndchance method Expected # of busts Actual # of Busts 12 84 (4/13)?=0.0947 7.95 11 13 61 (5/13)?=0.1479 9.02 13 14 67 (6/13)?=0.213 14.27 18 15 61 (7/13)?=0.2899 17.68 21 16 59 (8/13)?=0.3787 22.34 26 Total 332 71.26 89

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.

Summary

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%.

## My Response

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.

## Retest Option

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 Continues

Following is a letter I received from the Casino Bar attorneys on June 28, 2002:

Dear Sirs,

Re: COA World Entertainment Limited

Further to Prof. Yair Tauman's report, a leading Professor of Game Theory, PhD, at Tel Aviv University, we would like to draw your attention to the following:

Professor Yair Tauman is a renowned expert in the field of mathematics and statistics, with expertise in Game Theory and Probabilities. Prof. Tauman served as an associate editor of the 'International Journal of Game Theory' and the 'Games and Economic Behavior' publication. His teaching career spans over famous institutions in the US such as Stanford, Ohio State, SUNY at Stony Brook, Kellogg School of Business at Northwestern University and in the two largest universities in Israel, Tel Aviv and Jerusalem.

Prof. Tauman's finding prove, without dispute, that the Black Jack game played in Casino Bar is fair, leaving your conclusions with no basis and thereby untrue.

We reject the way your last report has been phrased and published. Pursuant to Prof. Tuaman's report, we expected to receive, to the very least, an apology. Accordingly, we are advised to instruct you to immediately rescind publications in this regard and post a fully pledged apology with respect to your reckless and defamatory implications as noted above, to be circulated via the same media that served your report.

Our client never turned to you to conduct a test of Casino Bar's Black Jack game, and still maintains that whether you perform a retest or otherwise is at your sole discretion.

In addition, we were instructed to assess COA's losses and damages resulting from your original posting and once these are crystallized, we would advice as to further measures to be taken against yourself and all parties involved in this matter. It is also to note, in this regard, that the losses and damages incurred by COA have intensified due to your inclusion of Casinoonair.com in your original report, in spite of the fact that you have clearly stated that you have not played at Casino on Air.

Nothing contained herein will be deemed to constitute a waiver of any of COA's rights or remedies, all of which are specifically reserved.

Best regards,

Abramovich, Yosef, Hakim

Following is my response sent by my attorney

Dear Mr. Levit:

As you know, I am an intellectual property attorney and appear on behalf of Michael Shackleford.

I am attaching Mr. Shackleford's response to your June 28 letter. I am in complete agreement with Mr. Shackleford's response on legal grounds.

Just because Mr. Tauman got a fair game during his tests does not mean Mr. Shackleford got a fair game on May 27, 2002, which is what Mr. Shackleford's report states. Casino Bar has access to the log files of Mr. Shackleford's play and has never denied Mr. Shackleford's hands that even your expert stated was "extremely unlikely" in a fair game. Further, it is also noted that destruction of the log files would have legal consequences, of which I am sure you are aware. Also, two of Mr. Shackleford's colleagues referenced in his report have irrefutably videotaped and taken screen shots of their results.

Since truth is a complete defense to defamation, you still have not pointed out which part of Mr. Shackleford's report is false. Please do so. If you cannot do so, Mr. Shackleford owes you no apology or retraction. Further it is noted that pursuing legal action would only cause great negative publicity for your client, because it would likely be a widely publicized case, and the irrefutable evidence discussed above would likely be introduced at trial and hence more widely publicized.

Sincerely,

{name removed online per attorney's request}, Esq.

(Mr. Shackleford's letter follows)

Dear Sirs:

In response to your last letter I am declining to make an apology on my web site or any significant changes. Following are my comments about the specific points of your letter.

As I said online I respect Professor Yair Tauman and his report. His credentials are not in dispute.

I do not claim on my site that Casino Bar always plays unfairly at blackjack. Rather I claim that I personally did not get a fair game on May 27. Even Mr. Tauman said "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." So your own expert is agreeing with me. Just because he got a fair game doesn't prove I didn't.

Regarding Casino on Air I am only reprinting data I got from another person who believes in honest Internet gaming. I do not claim to have personally played there. However I believe his results corroborate my own findings since Casino on Air is a sister casino to Casino Bar.

It does not matter that I wasn't asked to do the study. You or anyone advertising on my site open themselves to greater scrutiny. When I received several complaints from my readers about carrying Casino Bar ads I felt obligated to test the game myself.

You may tally up the damages as you see fit. Since I only am reporting truthfully what happened to me at Casino Bar I see no reason to feel accountable. It will be up to you to seek damages through the U.S. courts, in which I feel your chances of success are also "extremely unlikely.". Regards,

Michael Shackleford
June 29, 2002

## First Retest

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.

### Casino Bar Retest 1 Results

Dealer 2-card Total Bust on Third Card
Yes No Total
12 8 10 18
13 13 21 34
14 10 11 21
15 11 8 19
16 9 5 14
total 51 55 106
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.

### Retest 1 Expected Busts

Dealer 2-card Total Sample Total Probability of Bust Expected Busts
12 18 30.77% 5.54
13 34 38.46% 13.08
14 21 46.15% 9.69
15 19 53.85% 10.23
16 14 61.54% 8.62
total 106 47.15
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.

## Second Retest

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.

### Casino Bar Retest Results

Dealer 2-card Total Bust on Third Card
Yes No Total
12 6 25 31
13 5 27 32
14 10 26 36
15 14 19 33
16 8 20 28
total 43 117 160
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.

### Retest Expected Busts

Dealer 2-card Total Sample Total Probability of Bust Expected Busts
12 31 30.77% 9.54
13 32 38.46% 12.31
14 36 46.15% 16.62
15 33 53.85% 17.77
16 28 61.54% 17.23
total 160 73.46
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.