Black Jack Appendix 10
This appendix shall attempt to answer the question about the effect on the house edge of the continuous
shuffling machine (CSM). The continuous shuffling machine is a device that
randomly inserts discards back in the deck. With one in use it is like playing
against a freshly shuffled shoe every hand. This machine is not to be confused
with an automatic shuffler that shuffle and entire deck or shoe.
Although the CSM gets a lot of criticism from players the truth is that it
actually lowers the house edge. To prove this I ran almost 75 billion hands
through a simulation program. The following table displays the results according
to the rules and whether a cut card or CSM was used. The other rules were:
dealer stands on soft 17 (except single deck), double after a split allowed,
split to up to three hands, no surrender, no unusual rule variations.
| House Edge with and without CSM |
Number of Decks |
Cut Card |
CSM |
Difference |
| 1 |
0.164% |
0.051% |
0.113% |
| 2 |
0.264% |
0.201% |
0.063% |
| 4 |
0.407% |
0.373% |
0.034% |
| 6 |
0.454% |
0.434% |
0.020% |
| 8 |
0.477% |
0.463% |
0.014% |
The reason for the difference in results at all is difficult to explain. The
way the basic strategy player plays causes more large than small cards to be
played. When a specified number of hands is played per shoe the result is the
player friendly penetration into the deck. However the cut card removes that
effect and levels the playing field. The following two tables show the
distribution of ranks actually played in both a cut card and CSM games.
| Distribution of Ranks in Cut Card Game |
| Rank |
Number |
Expected |
Difference |
Chi-Squared |
| 1 |
85905301 |
85908934 |
-3633 |
0.15 |
| 2 |
85907560 |
85908934 |
-1374 |
0.02 |
| 3 |
85911516 |
85908934 |
2582 |
0.08 |
| 4 |
85901000 |
85908934 |
-7934 |
0.73 |
| 5 |
85902875 |
85908934 |
-6059 |
0.43 |
| 6 |
85906345 |
85908934 |
-2589 |
0.08 |
| 7 |
85904400 |
85908934 |
-4534 |
0.24 |
| 8 |
85912242 |
85908934 |
3308 |
0.13 |
| 9 |
85911202 |
85908934 |
2268 |
0.06 |
| 10 |
343653697 |
343635735 |
17962 |
0.94 |
| total |
1116816138 |
1116816138 |
0 |
2.86 |
| Distribution of Ranks in CSM Game |
| Rank |
Number |
Expected |
Difference |
Chi-Squared |
| 1 |
85906480 |
85879548 |
26932 |
8.45 |
| 2 |
85707548 |
85879548 |
-172000 |
344.48 |
| 3 |
85737570 |
85879548 |
-141978 |
234.72 |
| 4 |
85785213 |
85879548 |
-94335 |
103.62 |
| 5 |
85819356 |
85879548 |
-60192 |
42.19 |
| 6 |
85846280 |
85879548 |
-33268 |
12.89 |
| 7 |
85875012 |
85879548 |
-4536 |
0.24 |
| 8 |
85908944 |
85879548 |
29396 |
10.06 |
| 9 |
85930794 |
85879548 |
51246 |
30.58 |
| 10 |
343916926 |
343518192 |
398734 |
462.83 |
| Total |
1116434123 |
1116434123 |
0 |
1250.05 |
Note how the distribution is weighted towards large cards in the CSM game as
opposed to the even distribution in the cut card game. The Chi-Squared statistic
is a measurement of how far the results deviate from expected. The numbers in
the lower right cell of each table show a Chi-Squared statistic of less than 3
in the cut card game and 1250 in the CSM game.
However it must also be stressed that the CSM allows the dealer to deal
continuously, increasing the number of hands dealt per hour by about 20%. For
the basic strategy player this will result in a greater expected loss on an
hourly basis. In actual casinos I have only seen these machines used in shoe
games however the effect is the same on the Internet where it is common to see
games shuffled after every hand in all numbers of decks. The effect of shuffling
after every hand and the use of a CSM are the same.