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An Experimental Analysis of an Online Casino
This paper is an experiment and statistical analysis of one online casino,www.casinobar.com. It is structured as an independent peer review and experimental replication of a test conducted by Michael Shackleford (www.wizardofodds.com). The data in this study was collected from a 'real money' blackjack session conducted by Dan Pronovost at Casinobar.com on June 12, 2002.
The data in this study indicates that it is extremely unlikely (less than 0.014% probability) that the observed experimental data is the product of a normal game of blackjack without alteration of the dealt cards. Although this study makes no claims as to the actual cause of this statistical abnormality, the results are consistent with the practice of 'dealings seconds' exactly when the dealer has a hand total from 12 to 16, and the player has a hand total of 16 or more.
This study was conducted to provide independent third party analysis of one particular online casino, namelywww.casinobar.com. Shackleford conducted an experiment of www.casinobar.com and published the results in June 2002.
The purpose of this study is to replicate his experiment and confirm or deny his conclusions.
The hypothesis to be tested is as follows:
"Are the occurrences of the third dealt card to the dealer in blackjack atwww.casinobar.com consistent with expected probability in the case where the first two dealer cards total hard 12 to 16? This analysis is further broken down by classifying all such hands into two categories: when the player has a hand total less than 16, or 16 and greater."
To test this hypothesis, we conducted the following experiment:
Excluding an initial 45 rounds of blackjack played to perfect the data collection process, no further games or trials were played. No rounds are excluded from the collected data, nor was any other form of exclusion or intentional experimental bias introduced.
The table below shows the results from this trial for the hands matching the hypothesis:
Table 1: Experiment data from 500 hands of blackjack atwww.casinobar.com
The 'No bust' columns represent observed incidents of dealer hands where the third card dealt to the dealer did not cause the dealer to bust immediately. Note that this includes dealer hands where the dealer would later bust the hand (in four or more cards): the final state of the round is not relevant to this experiment. The 'Bust' columns represent dealer hands where the third dealt card caused the dealer have an immediate hard total greater than 21 (a bust).
We can easily compute the expected probability in a fair game of blackjack for a 'Bust' or 'No bust' result given any dealer hard hand combination. The following table summarizes these computed values (these are not experimental results):
Table 2: Expected probability for 'Bust' and 'No bust' dealer hands
Although you may be tempted to simply compare the observed proportions to these percentages, this would not be a useful statistical analysis. The proper statistical test is as follows:
"What is the probability of an experimental trial with the observed number of 'Bust' events or less in the total occurrences of each corresponding dealer hand."
This experiment can be analyzed using Bernoulli Trial methods. The probability of obtaining exactly x successes, each with probability p, in n trials is given by the following binomial distribution.
By summing this equation's values from 0 to the observed number of 'Bust' events, we derive the probability of the observed experimental results.
The value of the player hand should have absolutely no affect on the number of dealer 'No busts' in a fair game of blackjack (www.casinobar.com uses four decks and reshuffles after each round, eliminating any possible bias due to card exposure). We can properly test the hypothesis by analyzing the data in the two player hand classifications.
Analysis of Data
Using the above statistical approach, we can compute the probability of seeing the observed number of dealer 'Busts' or less (bold italic column):
Table 3: Probability of observed results
The four observed dealer busts with hard 16 when the player has a hand value of 16 or greater is very far from the expected 11.1. The probability of this happening naturally in a fair game of blackjack in our experiment is less than 0.08%, or once in 1250. The results for the remaining hard hands are all below expectation as well. Even more telling is the fact that with player hands of value less than 16, the number of busts meets or beats expectation in all cases. There is no statistical reason in a fair game of blackjack to expect that the player's hand total should affect the dealer's probability of busting on equivalent hands.
An alternative cumulative analysis can be completed by combining the total bust dealer hands. 28 were observed when the player hand total was greater than or equal to 16, and 45.5 were expected. Given the fixed and independent expected probabilities for each hard hand, the standard deviation can be computed and used to determine the overall likelihood of so few busts. Using this aggregate method, the probability of the total observed number of busts when the player hand total is 16 or greater is 0.0144% (once in 6944).
The data in this study indicates that it is extremely unlikely (less than 0.014% probability) that the observed experimental data is the product of a normal game of blackjack without alteration of the dealt cards. Further experimentation would be necessary to provide evidence for any actual hidden playing tactic being employed. But the observed results are consistent with the blackjack technique of 'dealing seconds', whereby the dealer avoids taking the next shoe card when they know it will bust their hand, and instead take the second-next shoe card. This practice would cause the statistically abnormal low number of busts seen in this experiment. The lack of abnormality in the dealer hands when the player hand is valued less than 16 shows that it is exceedingly likely that some form of bias is being employed selectively.
The data used in this study is available for peer review by other statisticians. A screen shot of each one of the 500 rounds of blackjack played was taken. A complete Excel spreadsheet showing detailed calculations is available online:www.deepnettech.com/casinobar_experiment.xls. For further information about this study, please contact Dan Pronovost at firstname.lastname@example.org.