## Effect of Number of Cards Remaining in the Game of Blackjack

Using combinatorial analysis we note the the overall expected value for a lesser number of decks is greater than that of more decks given the same set of rules using full shoes for comparison. For a number of decks very large this overall EV approaches the fixed value of an infinite shoe composition. An infinite shoe composition is one where there is a fixed probability of 1/13 of drawing any non-ten value rank and a fixed probability of drawing any ten value rank (T,J,Q,K) of 4/13. What we wish to explore is the overall expected value starting with a full shoe consisting of a given number of full decks dealt down to a number of cards that is less than the starting number of cards for this given number of decks.

### Single Deck Example

Shown here are the number of possible random compositions depending upon number of cards remaining to be dealt for 1 and 2 decks and the probabilities of them being randomly dealt. 1 deck will be used as an example. If 52 cards are present there is only 1 subset possible (the full shoe with probability of 1.) If 51 cards remain there are 10 subsets (1 of each rank can be removed.) If 50 cards remain there are 55 subsets (A-A, A-2, A-3, A-4, A-5, A-6, A-7, A-8, A-9, A-T, 2-2, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 2-T, 3-3, 3-4, 3-5, 3-6, 3-7, 3-8, 3-9, 3-T, 4-4, 4-5, 4-6, 4-7, 4-8, 4-9, 4-T, 5-5, 5-6, 5-7, 5-8, 5-9, 5-T, 6-6, 6-7, 6-8, 6-9, 6-T, 7-7, 7-8, 7-9, 7-T, 8-8, 8-9, 8-T, 9-9, 9-T, T-T can be removed.) For each differing number of cards remaining to be dealt there are a possible number of subsets/random probabilities. The maximum number of possible subsets from 1 deck is when 26 cards remain (1,868,755.) As an example these are the 715 possible 48 card compositions (2 through ace) and their probabilities listed in that order and separated by commas. Since all of the possiblities are known it should be possible to compute an overall expected value by combinatorial analysis by computing the overall EV for each subset and weight it by its probability, which I have done for 51, 50, 49, and 48 cards remaining. If basic strategy is used overall EV comes out to exactly the basic strategy EV for a full deck. I am sure that for any number of cards remaining this would prove to be true. This leads to the conclusion that varying only the number of cards remaining has no effect on a player using basic strategy.

### Effect of Requiring a Completed Round on Overall EV

Above it was shown for a random subset of a given number of cards a basic strategy player's expected value is exactly the full shoe basic strategy value. However, the only time overall EV is computable after the first round dealt from a full shoe is after a full round is completed. Let's again consider a 48 card subset from a single deck. Above it was shown that there are 715 random subsets and that (overall expected value for playing each with full shoe basic strategy) times (probability of each) = full shoe basic strategy EV. The complication is that in playing a round of blackjack (assuming 1 player versus dealer,) many if not most of the random 48 card subsets are not possible. For example the subset with 4 2's removed clearly cannot occur. In actuality a player's overall EV following a 4 card round would be very different than the value given 4 cards are randomly burned. It seems that early in a shoe that there would be a large variation of the actual subset probabilities from the random subset probabilites. In fact very early in the shoe many of the random subset probabilities would be 0 while others would be more probable. How much variation from random subset probabilities later in the shoe there is I couldn't say.

What we can say is that for a given number of cards remaining to be dealt:

1) basic strategy expected value is constant if all possible subset shoe compositions are possible
2) basic strategy expected value is variable if possible subsets are limited because of the requirement that a complete blackjack round must have been played before a new round can begin

Using an example of 1 player versus dealer dealt from a single deck let's explore case 2 a little more. A round must consist of a minimum of 4 cards. There is also a maximum number of cards. This maximum number would be dependent upon actual cards dealt and player strategy. (Dealer strategy is fixed.)

Whenever the finished round consists of the minimum number of 4 cards it is very most likely that the cards removed would consist mainly of high cards. This means that the following round would be played at an expected value of (much) less than the basic strategy EV of a full shoe (using basic strategy in both cases.)

Whenever the finished round consists of 5 cards expected value for the following round would very most likely be less than the basic strategy EV of a full shoe but more than the EV for the case when the round consisted of 4 cards.

As the number of cards consisting the finished round increases, EV using basic strategy would increase and at some number of cards exceed full shoe basic strategy. This is because a round consisting of more and more cards would necessarily be composed of more and more low cards, which would increases EV.

Therefore the theory is that in the absence of a cut card effect a player using basic strategy will be playing at basic strategy EV over all rounds absent a floating advantage (below.)

This looks at floating advantage using random subsets as a model. As was shown above basic strategy EV for a random deal is fixed whereas actually subsets are not completely random but are dependent on the requirement of a finished blackjack round, where basic strategy EV is variable. Random subsets consist of all possible subsets whereas requiring a finished round limits which subsets are possible.

The tendency for increased overall expected value for a lesser number of cards remaining is known as floating advantage. There is no gain in EV for just reducing the number of cards remaining. However, we may wish to consider what may happen as cards are dealt AND a neutral shoe composition is present where a neutral shoe is defined as a full shoe composition. The problem is, "How would we pick neutral shoe compositions out of all of those possible?" The answer is to use some sort of counting system. Example counting systems of HiLo -1 {2,3,4,5,6}, 0 {7,8,9}, +1 {T,A} and Wong Halves (doubled) -1 {2,7}, -2 {3,4,6} -3 {5}, 0 {8}, +1 {9} +2 {T,A} will be used. For both HiLo and Wong Halves a neutral count is defined as a running count of 0.

Before we start we note that as cards are dealt overall EV can possibly be affected by a cut card effect. We can eliminate the possibility of any cut card effect by insisting that the last round that is played in a shoe is the first round where it is possible that a cut card will be encountered. If a basic strategy player is able to follow this policy his overall expected value will be the same as the overall basic strategy EV for whatever the starting number of decks is. Basically there is no cut card effect as long as the cut card cannot be encountered. There is also no cut card effect on the first round that the cut card may be encountered. However, if the cut card has not appeared on the above round this is when the cut card effect is manifested. This is seen in these simulations.

We will approximate the overall EV when 26 cards remain and the count is neutral and compare it to the overall EV for a full deck using HiLo and Wong Halves (doubled) as the counts which determine the neutrality of the 26 card 1/2 deck composition. For both HiLo and Wong Halves a neutral count is defined as a running count of 0. I have written a program that computes the probability of drawing each rank given a running count and number of cards remaining. Below is the output for HiLo and Wong Halves (doubled) for 26 cards remaining and a ruunning count of 0 for these 2 counting systems.

```HiLo
Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Subgroup removals: None
Specific removals (1 - 10): {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Number of subsets for above conditions: 7
Prob of running count 0 with above removals from 1 deck: 0.124165

p[1] 0.0769231  p[2] 0.0769231  p[3] 0.0769231  p[4] 0.0769231  p[5] 0.0769231
p[6] 0.0769231  p[7] 0.0769231  p[8] 0.0769231  p[9] 0.0769231  p[10] 0.307692

Press x or X to exit program (it may take some time to close,)
any other key to enter more data for same count tags/decks:

Wong Halves (doubled)
Count tags {2,-1,-2,-2,-3,-2,-1,0,1,2}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Subgroup removals: None
Specific removals (1 - 10): {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Number of subsets for above conditions: 281
Prob of running count 0 with above removals from 1 deck: 0.0591537

p[1] 0.0769231  p[2] 0.0769231  p[3] 0.0769231  p[4] 0.0769231  p[5] 0.0769231
p[6] 0.0769231  p[7] 0.0769231  p[8] 0.0769231  p[9] 0.0769231  p[10] 0.307692

Press x or X to exit program (it may take some time to close,)
any other key to enter more data for same count tags/decks:
```

As you can see the probabilities for drawing each rank is exactly the same for a running count of 0 when 26 cards remain from a randomly dealt single deck for HiLo and Wong Halves. Additionally the probability of each non-ten rank is 1/13 and the probability of a ten value card is 4/13. This exactly matches the probabilites for a full deck. It turns out that for any counting system and any number of decks a neutral count when half of a shoe remains given a random deal, this is true. There are 7 possible HiLo subsets for 26 cards remaining and a running count of 0 whereas for Wong Halves there are 281. Each of these subsets has a probability of occurrence. The final rank probabilities are obtained by weighting the probabilities of each rank in each subset by the probability of the subset. Since the probabilities of each rank for half of a deck exactly match those of a full deck, we will use combinatorial analysis for the 26 card subset of {2,2,2,2,2,2,2,2,2,8} to approximate overall EV for a 1/2 deck running count of 0 dealt from a full single deck with rules of dealer stands on soft 17, double on any 2 cards, 1 split allowed on any pair, 1 card drawn to split aces, no doubling after splitting, no surrender. Below are the overall expected values for this representative subset.

```full single deck EV (optimal composition dependent): +.0248%
full single deck EV (total dependent): -.0147%
1/2 single deck EV (optimal composition dependent): +.8712%
1/2 single deck EV (using composition dependent basic strategy): +.8105%
1/2 single deck EV (using total dependent basic strategy): +.6932%
(composition dependent splits are computed using the optimal strategy of the first split hand to determine
both optimal and (full shoe) basic strategies and resulting expected values)
```

From the above data it appears that there at least tends to be a floating advantage for a neutral shoe (as determined by a counting system) for less than a full shoe. We are using the above {2,2,2,2,2,2,2,2,2,8} subset to approximate EV when in actuality there are 7 possible 26 card HiLo subsets dealt from a single deck that have a running count of 0

```7{2,3,4,5,6}   12{7,8,9}  7{T,A}   probability 0.000012117616101733498
8{2,3,4,5,6}   10{7,8,9}  8{T,A}   probability 0.0021118732812302411
9{2,3,4,5,6}   8{7,8,9}   9{T,A}   probability 0.028158310416403244
10{2,3,4,5,6}  6{7,8,9}   10{T,A}  probability 0.063600237127182713
11{2,3,4,5,6}  4{7,8,9}   11{T,A}  probability 0.028158310416403199
12{2,3,4,5,6}  2{7,8,9}   12{T,A}  probability 0.0021118732812302411
13{2,3,4,5,6}  0{7,8,9}   13{T,A}  probability 0.000012117616101733498
```

These total to approximately 0.124165 which is the probability of a randomly dealt 26 card subset from a single deck with a HiLo running count of 0. As is stated above the final rank probabilities are obtained by weighting the probabilities of each rank in each subset by the probability of the subset and turn out to be 1/13 for each non-ten rank and 4/13 for a ten value card. For Wong Halves there are 281 such subsets which total to 0.0591537 so there is less probability of a 0 running count for Wong Halves than HiLo. Rather than using the above representative subset the EV for each individual subset could be simmed using basic strategy, multipled by prob(subset)/prob(RC 0), and summed to get the basic strategy EV for a 26 card subset with a running count of 0 dealt from a single deck. I would be surprised if it differed much from basic strategy EV of the representative subset which can be computed using combinatorial analysis.

The above example is for a single deck dealt such that 1/2 deck remains with a neutral count. The same method of dealing half of the starting number of cards with a condition of a neutral count would be applicable to any number of starting decks. The greater the number of starting decks the less is the gain in overall expected value for a half shoe as compared to a full shoe.

### 52 Card Neutral HiLo Count Dealt From 6 Decks

```Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 6
Cards remaining: 52
Initial running count (full shoe): 0
Running count: 0
Subgroup removals: None
Specific removals (1 - 10): {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Number of subsets for above conditions: 27
Prob of running count 0 with above removals from 6 decks: 0.0687657

p[1] 0.0767745  p[2] 0.0767745  p[3] 0.0767745  p[4] 0.0767745  p[5] 0.0767745
p[6] 0.0767745  p[7] 0.0774182  p[8] 0.0774182  p[9] 0.0774182  p[10] 0.307098

Press x or X to exit program (it may take some time to close,)
any other key to enter more data for same count tags/decks:
```

6 decks dealt to a point where 52 cards remain with a 0 HiLo running count results in 27 possible subsets. If the above rank probabilities are multiplied by 52, the resulting 52 card composition would be {3.992274, 3.992274, 3.992274, 3.992274, 3.992274, 3.992274, 4.0257464, 4.0257464, 4.0257464, 15.969096} (Ace through ten.) The weighted shoe composition cannot be expressed in whole numbers but it is pretty close to a composition of {4,4,4,4,4,4,4,4,4,16} which is the composition of a single deck. The expected value for a single deck with rules of dealer stands on soft 17, double on any 2 cards, 1 split allowed on any pair, 1 card drawn to split aces, no doubling after splitting, no surrender is +.0248%. The expected value for 6-deck composition dependent basic strategy played versus this single deck composition is -.0128% for the same rules. The full shoe 6-deck EV is -.5788%. This is made as just an observation with no conclusion.

## Why would expected value tend to be higher when there are fewer cards and a neutral deck?

In order to explore this question we will turn to another example. We will compare EV for all 2-card hands for 1 full deck and 100 full decks with rules of dealer stands on soft 17, double on any 2 cards, 1 split allowed on any pair, 1 card drawn to split aces, no doubling after splitting, no surrender. (These are composition dependent values.)

```				(Expected Values are in Percent)
Hand	Decks	Probability	Exp Value	Overall EV	Difference	Dlr bust

A-A	1	0.004524887	40.55		0.183484163			.3013
100	0.005903502	30.87		0.182241111	-.001243052	.2818

A-2	1	0.012066365	-.5364		-0.006472398			.2918
100	0.011836596	-2.808		-0.033237161	-.026764763	.2817

A-3	1	0.012066365	-3.069		-.0370316742			.2921
100	0.011836596	-5.509		-.0652078074	-.028176133	.2817

A-4	1	0.012066365	-6.972		-.0841266968			.2921
100	0.011836596	-8.107		-.0959592838	-.011832587	.2817

A-5	1	0.012066365	-10.80		-.1303167420			.2924
100	0.011836596	-10.50		-.124284258	+.006032484	.2817

A-6	1	0.012066365	-5.734		-.0691885369			.2903
100	0.011836596	-7.856		-.0929882982	-.0237997613	.2817

A-7	1	0.012066365	5.222		.06301055803			.2947
100	0.011836596	3.151		.03729711310	-.02571344493	.2817

A-8	1	0.012066365	27.49		.33170437385			.2935
100	0.011836596	26.60		.3148534536	-.01685092025	.2817

A-9	1	0.012066365	59.19		.71420814435			.2915
100	0.011836596	57.97		.68616747012	-.02804067423	.2817

A-T	1	0.048265460	144,5		6.97435897			.2901
100	0.047346383	142.9		6.7657981307	-.2085608393	.2817

2-2	1	0.004524887	-17.00		-0.076923079			.2825
100	0.005903502	-18.83		-0.11116294266	-0.03423986366	.2816

2-3	1	0.012066365	-21.71		-0.261960784			.2829
100	0.011836596	-21.92		-0.25945818432	+.00250259968	.2816

2-4	1	0.012066365	-24.68		-0.297797888			.2830
100	0.011836596	-23.88		-0.28265791248	+.01513997552	.2816

2-5	1	0.012066365	-21.29		-0.256892911			.2827
100	0.011836596	-21.38		-0.25306642248	+.00382648852	.2816

2-6	1	0.012066365	-12.57		-.15167420805			.2811
100	0.011836596	-12.69		-.15020640324	+.006032484	.2816

2-7	1	0.012066365	1.297		.015650075405			.2855
100	0.011836596	-.5246		-.006209478262	+.00146780481	.2816

2-8	1	0.012066365	22.75		.27450980375			.2841
100	0.011836596	20.60		.2438338776	-.03067592615	.2816

2-9	1	0.012066365	30.76		.3711613874			.2821
100	0.011836596	30.96		.36646101216	-.00470037524	.2816

2-T	1	0.048265460	-30.23		-1.4590648558			.2811
100	0.047346383	-31.59		-1.49567223897	-.03660738317	.2816

3-3	1	0.004524887	-21.87		-0.09895927869			.2832
100	0.005903502	-22.65		-0.1337143203	-0.03475504161	.2816

3-4	1	0.012066365	-22.44		-0.2707692306			.2828
100	0.011836596	-21.39		-0.25318478844	+0.01758444216	.2816

3-5	1	0.012066365	-12.08		-0.1457616892			.2830
100	0.011836596	-12.69		-0.15020640324	-0.00444471404	.2816

3-6	1	0.012066365	2.093		0.025254901945			.2814
100	0.011836596	-.5164		-0.006112418174	-0.031367320119	.2816

3-7	1	0.012066365	23.58		0.2845248867			.2858
100	0.011836596	20.61		0.24395224356	-0.04057264314	.2816

3-8	1	0.012066365	32.24		0.3890196076			.2844
100	0.011836596	30.97		0.36657937812	-0.02244022948	.2816

3-9	1	0.012066365	-33.04		-0.3986726996			.2828
100	0.011836596	-31.62		-0.37427316552	+0.02439953408	.2816

3-T	1	0.048265460	-33.56		-1.6197888376			.2814
100	0.047346383	-34.74		-1.64481334542	-0.02502450782	.2816

4-4	1	0.004524887	-11.25		-0.05090497875			.2829
100	0.005903502	-12.68		-0.07485640536	-0.02395142661	.2816

4-5	1	0.012066365	2.726		0.03289291099			.2831
100	0.011836596	-.5093		-0.006028378343	-0.038921289333	.2816

4-6	1	0.012066365	24.29		0.29309200585			.2814
100	0.011836596	20.61		0.24395224356	-0.04913976229	.2816

4-7	1	0.012066365	33.41		0.40313725465			.2858
100	0.011836596	30.98		0.36669774408	-0.03643951057	.2816

4-8	1	0.012066365	-32.78		-0.3955354447			.2849
100	0.011836596	-31.62		-0.37427316552	0.02126227918	.2816

4-9	1	0.012066365	-33.77		-0.40748114605			.2828
100	0.011836596	-34.74		-0.41120334504	-0.00372219899	.2816

4-T	1	0.048265460	-36.55		-1.764102563			.2815
100	0.047346383	-37.25		-1.76365276675	0.00044979625	.2816

5-5	1	0.004524887	25.03		0.11325792161			.2833
100	0.005903502	20.62		0.12173021124	0.00847228963	.2816

5-6	1	0.012066365	34.43		0.41544494695			.2817
100	0.011836596	30.99		0.36681611004	-0.04862883691	.2816

5-7	1	0.012066365	-32.63		-0.39372548995			.2865
100	0.011836596	-31.61		-0.37415479956	0.01957069039	.2816

5-8	1	0.012066365	-36.16		-0.4363197584			.2852
100	0.011836596	-34.76		-0.41144007696	0.02487968144	.2816

5-9	1	0.012066365	-36.80		-0.444042232			.2831
100	0.011836596	-37.25		-0.440913201	0.003129031	.2816

5-T	1	0.048265460	-39.44		-1.9035897424			.2817
100	0.047346383	-39.58		-1.87396983914	0.02961990326	.2816

6-6	1	0.004524887	-28.95		-0.13099547865			.2806
100	0.005903502	-29.22		-0.17250032844	-0.04150484979	.2816

6-7	1	0.012066365	-36.49		-0.44030165885			.2848
100	0.011836596	-34.76		-0.41144007696	0.02886158189	.2816

6-8	1	0.012066365	-37.31		-0.45019607815			.2834
100	0.011836596	-37.25		-0.440913201	0.00928287715	.2816

6-9	1	0.012066365	-39.85		-0.48084464525			.2813
100	0.011836596	-39.58		-0.46849246968	0.01235217557	.2816

6-T	1	0.048265460	-40.06		-1.9335143276			.2800
100	0.047346383	-41.72		-1.97529109876	-0.04177677116	.2816

7-7	1	0.004524887	-31.47		-0.14239819389			.2889
100	0.005903502	-30.75		-0.1815326865	-0.03913449261	.2817

7-8	1	0.012066365	-37.58		-0.4534539967			.2874
100	0.011836596	-39.56		-0.46825573776	-0.01480174106	.2816

7-9	1	0.012066365	-40.01		-0.48277526365			.2854
100	0.011836596	-41.72		-0.49382278512	-0.01104752147	.2816

7-T	1	0.048265460	-29.23		-1.4107993958			.2836
100	0.047346383	-29.17		-1.38109399211	0.02970540369	.2816

8-8	1	0.004524887	-17.83		-0.13923077299			.2861
100	0.005903502	-20.62		-0.17480269422	-0.03557192123	.2816

8-9	1	0.012066365	-28.26		-0.4534539967			.2836
100	0.011836596	-29.16		-0.46825573776	-0.01480174106	.2816

8-T	1	0.048265460	-.3315		-1.4107993958			.2822
100	0.047346383	-.7027		-1.38109399211	0.02970540369	.2816

9-9	1	0.004524887	5.113		0.023135747231			.2815
100	0.005903502	3.601		0.021258510702	-0.001877236529	.2816

9-T	1	0.048265460	27.55		1.329713423			.2802
100	0.047346383	26.60		1.2594137878	-0.0702996352	.2816

T-T	1	0.0904977376 	58.71		5.313122174496			.2788
100	0.0946335834	57.97		5.485908829698	0.172786655202	.2816
```

We see that there are some hands dealt from 100 decks that have a greater expected value than the same hand dealt from a single deck. However, on balance EV tends to be higher for a single deck. The overall EV for a single deck is +.0248% whereas for 100 decks it is -.6838%. It seems that due to the dynamics of the probabilities for fewer cards that overall EV tends to be higher for fewer cards at least for shoe compositions that are not too unreasonably extreme.