Blackjack Appendix 10

Last update: March 19, 2006

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, which shuffles 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 large simulations both with and without a cut card under typical blackjack rules. The following table shows how much the house advantage is reduced by the use of a continuous shuffler compared to a cut card game.

CSM reduction in House Edge
Number of Decks	Reduction
1	0.113%
2	0.063%
4	0.034%
5	0.028%
6	0.020%
8	0.014%

The next two tables show the distribution of each rank in both types of games, both using a single deck.

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.

I have been asked if more small cards come out in a cut card game when can they be expected to appear. There is no particular time. The last hand in a cut card has just about the same odds as the overall shoe. However if the dealer deals out much more than the average number of hands in a cut card game the last hands tend to be very bad for the player. This is because in the early hands the players and dealer didn't hit much, which in turn is because lots of large cards came out, leaving more small cards for later in the shoe. So if you notice that you have already passed the average number of hands and the cut card is still a ways off then the deck is probably small card rich and it would be a good time to drop your bet or take a walk. However with other players jumping in and out of the game, and inconsistent cut card placement the practicality of this strategy is very small.

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.

Links

Outstanding article on the cut card effect at BlackjackInColor.com.

There is a good discussion of the cut card effect at Casino Verite.

Various posts about the cut card effect at bjmath.com.

The Wizard's other sites: Wizard of Vegas, Wizard of Macau, Math Problems The Wizard recommends: The Bear Growls, Casinomeister, Online Casino Suite