Blackjack: Cards that count.
Cards that can count: A session of 104 single handed, Blackjack hands played online on 23rd August was analysed. The chart below shows the distribution of cards at various counts for both the Dealer and the Player. The bias in the value of cards drawn by the Dealer is obvious from top right to bottom left. The value of the cards clearly becomes smaller as the count increases. No such bias is evident in the Players cards at the bottom of the chart.
The relevant spreadsheets are attached.
Refer to attached *.xlsx spreadsheet, sheet 2.
A chi squared test on the total of the Dealers cards returns a result of 0.976, pretty close to a perfect fit, while a similar result for the Players cards is 0.702. So statistically the Player cards are the less good fit yet the bias in the upper half of the attached chart is evident to the eye. On a count of 14 the Dealers average card value was 7, on a count of 15 it was 5 and, on 16, it was 4. Pretty smart.
The following is taken from a Session of 520 hands on 25th August. The Chi Squared results for this sample are below. The first figure in a pair is the number of cards the second figure is the shi squared result.
.All .Dealt Drawn
Dealer 1450 0.115 1040 0.300 412 0.149
Player 1373 0.012 1040 0.334 335 0.002
Total 2823 0.043 2080 0.314 747 0.048
All these results are well below an average fit and the result of 0.043 for a total of 2,823 cards needs some explanation. The results for the Players Drawn cards are especially bad. The results returned are actually worse than removing all the Aces, Kings and Queens from a deck and replacing them with 2s, 3s and 4s. It is even more surprising when continuous shuffling should, theoretically, minimise variance because the impact of discards is minimised. The only results that look acceptable are those for the first two cards dealt; to me the remainder look suspect.
Here are the drawn cards that gave the result of 0.0026 (1:400) from the table above:
Ace K Q J 10 9 8 7 6 5 4 3 2 Total
21 28 24 34 21 31 20 21 11 41 20 35 28 335
The attached Spreadsheet page 3 show the detail.
The Player drew on a count of 15 in 33 hands, won 3 and was dealt twelve 3s and eight Queens. The Chi Squared result for this distribution is 3.81E-08 or, odds of 0.0000000381 or 1: 26,315,700. Clearly, whatever controls are in place to ensure a fair game are so broad as to be meaningless. These figures are undoubtedly subsumed into a total for the player or for all players by which time their significance disappears.
Without access to the source code (and the ability to make sense of it) it is impossible to say what is happening but it is impossible to believe that this is random. The totals may look fine but the details look anything but fine to me. If any part of the game is discretionary and not random then the game is no longer fair and should not be called Blackjack.
There are people on this board with experience of online systems and fair gaming audits. Would they care to comment ?
Cheers
CGB
