MJ: Cards used per player per hand

MJ · 01-12-2009, 05:36 PM

In a heads up game, on average, how many cards are used per player per hand?

Wouldn't this number change with different rules? LS would probably decrease this avg by a bit b/c fewer hits would means fewer cards expended from the shoe. Conversely, RSA would probably increase this number by a bit.

The conventional number is 2.7 cards used per player, but I believe this assumes more than 1 player at the table. But when heads up, the dealer does not have to take additional hit cards when the player busts, which leads me to believe the avg should be smaller than 2.7 cards per player.

The rules pertaining to my game are 6D, S17, DAS, DA2, heads up.

MJ

Dog Hand · 01-12-2009, 08:53 PM

MJ,

I simmed your game twice in CVData: the first time with LS, and the second time without it. Both sets of output are shown below.

If we confine our observations to only the first 38 rounds of each shoe (after that point the cut card starts to have a significant impact), we can see on average the LS game uses 2.711 cards per hand per round, while the no LS game uses 2.733 cards per hand per round: a difference of 0.022 cards per hand per round.

Thus, your assumption that the average is changed by LS is indeed correct.

Dog Hand

MJ cards per hand

Monday, January 12, 2009, 09:36 PM

Configuration

[B]

Rounds per shoe	0
Burn cards	1
Shuffle point	78
Decks	6
Hands per hour	100
Players	1
Players come and go	0
Your Seat	1

Rules

Double down after split Soft DD counts as Hard Hand Late Surrender
No Ace splits Resplit Aces Dealer hits soft 17
Cards dealt face down Dealer shows burn cards Multiple draws after Ace split
Dealer peeks on 10 Dealer peeks on Ace Insurance
No dealer hole card dealt Early Surrender vs. 10 Split tens must be same value
Burn card after each round Double Down Rescue Multi-Card Surrender
Early Surrender vs. Ace Insure Blackjack Only Discard Split
Discard Double Bottom card dealt Double Down Rescue
Multi-Card Surrender Early Surrender vs. Ace AJ Blackjack pays 0:10
AJ Spades pays 0:10 AJ Black pays 0:10 AJ Suited pays 0:10
Suited BJ pays 0:10 Spades BJ pays 0:10 Five cards unbusted pays 0:10
Six cards unbusted pays 0:10 Five card 21 pays 0:10 Six card 21 pays 0:10
777 pays 0:10 Suited 777 pays 0:10 678 pays 0:10
Suited 678 pays 0:10 Bonus must win to pay Blackjack pays 15:10
Tied Blackjack Wins Tied Blackjack Loses Dealer wins tied 17
Dealer wins tied 18 Dealer wins tied 19 Dealer wins tied 20
Dealer wins tied 21 DD on Any hard hand DD on Any soft hand
Split to 4 hands DD on 2 cards Blackjack Wins All
Original Bets && Busted (OBBO) Original Bets Only (OBO) Busted Bets + 1 (BB+1)

Here, results are grouped by hand number. The average depth of each hand in a round is shown by cards and percentage of the shoe. Hands won, lost and tied is provided for each level as well as the total units bet, average bet and total won or lost. When using an effective strategy, the amount bet and won should increase as the depth increases. Action is Total dollars bet per hour and Win Rate is dollars won per hour.

Player #1: Hand Depth
Avg. Depth Hands Units Dollars per Hour Average
Hand Cards Shoe Won Lost Tied Total Bet Avg Bet Won/Lost Action Win Rate RC TC
1 1.00 0.32% 4,028,688 4,653,982 775,273 10,420,730 1.130 -37,159 $260.52 ($0.93) 0.0 0.0
2 6.41 2.05% 4,030,750 4,654,360 774,057 10,427,490 1.131 -34,561 $260.69 ($0.86) 0.0 0.0
3 11.81 3.79% 4,035,344 4,649,597 775,941 10,438,980 1.132 -28,556 $260.97 ($0.71) 0.0 0.0
4 17.22 5.52% 4,036,564 4,649,441 775,466 10,462,850 1.135 -33,206 $261.57 ($0.83) 0.0 0.0
5 22.63 7.25% 4,038,576 4,646,159 777,077 10,504,020 1.139 -30,767 $262.60 ($0.77) 0.0 0.0
6 28.04 8.99% 4,038,288 4,646,314 777,910 10,559,000 1.145 -32,020 $263.97 ($0.80) 0.0 0.0
7 33.46 10.72% 4,042,660 4,643,081 777,365 10,662,010 1.156 -23,946 $266.55 ($0.60) 0.0 0.0
8 38.87 12.46% 4,043,693 4,641,963 778,688 10,888,040 1.181 -31,542 $272.20 ($0.79) 0.0 0.0
9 44.29 14.19% 4,047,619 4,635,827 781,296 11,034,830 1.197 -17,179 $275.87 ($0.43) 0.0 0.0
10 49.71 15.93% 4,049,417 4,634,648 782,351 11,178,040 1.212 -14,113 $279.45 ($0.35) 0.0 0.0
11 55.12 17.67% 4,046,977 4,635,719 782,192 11,338,030 1.230 -22,078 $283.45 ($0.55) 0.0 0.0
12 60.54 19.40% 4,049,128 4,632,169 783,519 11,662,170 1.265 -5,852 $291.55 ($0.15) 0.0 0.0
13 65.96 21.14% 4,044,839 4,634,922 784,539 12,105,560 1.313 -14,416 $302.64 ($0.36) -0.1 0.0
14 71.38 22.88% 4,050,719 4,630,432 784,029 12,336,000 1.338 9,361 $308.40 $0.23 -0.1 0.0
15 76.80 24.62% 4,047,137 4,634,751 784,578 12,529,640 1.359 3,196 $313.24 $0.08 -0.1 0.0
16 82.22 26.35% 4,046,220 4,632,343 787,355 12,856,050 1.394 2,238 $321.40 $0.06 -0.1 0.0
17 87.64 28.09% 4,049,063 4,628,979 789,455 13,716,020 1.488 25,450 $342.90 $0.64 -0.1 0.0
18 93.06 29.83% 4,048,499 4,628,415 790,532 14,577,370 1.581 27,755 $364.43 $0.69 -0.1 0.0
19 98.48 31.57% 4,050,468 4,624,874 792,433 14,870,150 1.613 37,057 $371.75 $0.93 -0.1 0.0
20 103.91 33.30% 4,053,765 4,622,067 791,949 15,103,500 1.638 54,137 $377.59 $1.35 -0.1 0.0
21 109.33 35.04% 4,054,176 4,623,050 791,831 15,583,110 1.690 60,024 $389.58 $1.50 -0.1 0.0
22 114.75 36.78% 4,049,961 4,626,064 795,137 16,695,200 1.811 82,112 $417.38 $2.05 -0.1 0.0
23 120.18 38.52% 4,053,819 4,619,844 796,161 17,579,950 1.907 116,548 $439.50 $2.91 -0.1 0.0
24 125.60 40.26% 4,051,949 4,622,581 795,473 17,845,790 1.936 107,729 $446.14 $2.69 -0.1 0.0
25 131.02 41.99% 4,053,655 4,620,767 796,782 18,108,540 1.964 116,522 $452.71 $2.91 -0.1 0.0
26 136.45 43.73% 4,053,127 4,620,459 800,624 19,057,140 2.067 143,440 $476.43 $3.59 -0.1 0.0
27 141.87 45.47% 4,053,196 4,620,038 803,506 21,098,060 2.288 185,925 $527.45 $4.65 -0.1 0.0
28 147.30 47.21% 4,056,182 4,614,428 806,241 22,514,010 2.442 203,935 $562.85 $5.10 -0.1 0.0
29 152.72 48.95% 4,056,531 4,614,220 807,633 22,857,290 2.479 230,645 $571.43 $5.77 -0.1 0.0
30 158.15 50.69% 4,061,099 4,609,385 809,971 23,304,950 2.528 289,419 $582.62 $7.24 -0.1 0.0
31 163.58 52.43% 4,061,327 4,611,408 808,904 24,011,650 2.604 288,773 $600.29 $7.22 -0.1 0.0
32 169.00 54.17% 4,059,352 4,615,857 808,944 24,748,860 2.684 303,234 $618.72 $7.58 -0.1 0.0
33 174.43 55.91% 4,059,648 4,614,317 810,888 25,918,360 2.811 321,146 $647.96 $8.03 -0.1 0.0
34 179.86 57.65% 4,064,795 4,611,641 810,401 27,202,840 2.950 387,781 $680.07 $9.69 -0.1 0.0
35 185.28 59.39% 4,062,906 4,610,523 815,306 28,316,470 3.071 433,087 $707.91 $10.83 -0.1 0.0
36 190.71 61.13% 4,063,901 4,610,482 818,221 29,678,280 3.219 448,379 $741.96 $11.21 -0.1 0.0
37 196.14 62.87% 4,068,889 4,607,797 819,724 31,071,560 3.370 539,508 $776.79 $13.49 -0.1 0.0
38 201.57 64.61% 4,068,464 4,609,119 822,915 32,968,860 3.576 558,277 $824.22 $13.96 -0.1 0.0
39 207.00 66.35% 4,071,797 4,605,430 825,722 34,672,470 3.761 679,641 $866.81 $16.99 -0.1 0.0
40 212.42 68.08% 4,069,796 4,603,396 826,676 35,689,400 3.874 679,930 $892.23 $17.00 -0.1 0.0
41 217.75 69.79% 4,045,433 4,576,184 823,732 36,226,860 3.956 754,373 $905.67 $18.86 -0.2 0.0
42 222.66 71.37% 3,884,775 4,392,650 787,788 34,611,030 3.938 705,047 $865.28 $17.63 -0.3 -0.1
43 226.56 72.62% 3,234,874 3,660,857 656,505 27,415,650 3.743 545,255 $685.39 $13.63 -0.8 -0.4
44 229.12 73.44% 1,903,225 2,152,830 383,681 14,314,330 3.322 245,416 $357.86 $6.14 -1.7 -0.9
45 230.61 73.91% 622,040 703,238 124,806 3,873,389 2.749 47,192 $96.83 $1.18 -2.9 -1.7
46 231.47 74.19% 93,547 104,866 18,424 462,786 2.194 6,447 $11.57 $0.16 -4.5 -2.8
47 231.98 74.35% 5,515 6,234 1,081 22,438 1.801 -39 $0.56 $0.00 -6.3 -3.9
48 232.31 74.46% 128 140 17 417 1.504 -35 $0.01 $0.00 -8.0 -5.1

************************************************** *********

MJ cards per hand

Monday, January 12, 2009, 09:46 PM

Configuration

[B]

Rounds per shoe	0
Burn cards	1
Shuffle point	78
Decks	6
Hands per hour	100
Players	1
Players come and go	0
Your Seat	1

Rules

Double down after split Soft DD counts as Hard Hand Late Surrender
No Ace splits Resplit Aces Dealer hits soft 17
Cards dealt face down Dealer shows burn cards Multiple draws after Ace split
Dealer peeks on 10 Dealer peeks on Ace Insurance
No dealer hole card dealt Early Surrender vs. 10 Split tens must be same value
Burn card after each round Double Down Rescue Multi-Card Surrender
Early Surrender vs. Ace Insure Blackjack Only Discard Split
Discard Double Bottom card dealt Double Down Rescue
Multi-Card Surrender Early Surrender vs. Ace AJ Blackjack pays 0:10
AJ Spades pays 0:10 AJ Black pays 0:10 AJ Suited pays 0:10
Suited BJ pays 0:10 Spades BJ pays 0:10 Five cards unbusted pays 0:10
Six cards unbusted pays 0:10 Five card 21 pays 0:10 Six card 21 pays 0:10
777 pays 0:10 Suited 777 pays 0:10 678 pays 0:10
Suited 678 pays 0:10 Bonus must win to pay Blackjack pays 15:10
Tied Blackjack Wins Tied Blackjack Loses Dealer wins tied 17
Dealer wins tied 18 Dealer wins tied 19 Dealer wins tied 20
Dealer wins tied 21 DD on Any hard hand DD on Any soft hand
Split to 4 hands DD on 2 cards Blackjack Wins All
Original Bets && Busted (OBBO) Original Bets Only (OBO) Busted Bets + 1 (BB+1)

Here, results are grouped by hand number. The average depth of each hand in a round is shown by cards and percentage of the shoe. Hands won, lost and tied is provided for each level as well as the total units bet, average bet and total won or lost. When using an effective strategy, the amount bet and won should increase as the depth increases. Action is Total dollars bet per hour and Win Rate is dollars won per hour.

Player #1: Hand Depth
Avg. Depth Hands Units Dollars per Hour Average
Hand Cards Shoe Won Lost Tied Total Bet Avg Bet Won/Lost Action Win Rate RC TC
1 1.00 0.32% 4,168,957 4,585,134 797,943 10,527,530 1.133 -37,685 $263.19 ($0.94) 0.0 0.0
2 6.46 2.07% 4,169,486 4,583,296 798,974 10,530,690 1.133 -36,237 $263.27 ($0.91) 0.0 0.0
3 11.91 3.82% 4,168,091 4,583,767 801,425 10,543,530 1.135 -37,561 $263.59 ($0.94) 0.0 0.0
4 17.37 5.57% 4,167,858 4,584,263 800,498 10,563,430 1.137 -37,697 $264.09 ($0.94) 0.0 0.0
5 22.83 7.32% 4,167,213 4,584,837 801,270 10,604,110 1.141 -39,769 $265.10 ($0.99) 0.0 0.0
6 28.29 9.07% 4,166,998 4,584,684 800,470 10,662,160 1.147 -38,925 $266.55 ($0.97) 0.0 0.0
7 33.75 10.82% 4,167,709 4,582,590 802,445 10,768,650 1.159 -31,791 $269.22 ($0.79) 0.0 0.0
8 39.21 12.57% 4,169,257 4,578,953 804,883 11,000,960 1.184 -23,680 $275.02 ($0.59) 0.0 0.0
9 44.67 14.32% 4,165,504 4,585,075 803,142 11,147,140 1.199 -35,813 $278.68 ($0.90) 0.0 0.0
10 50.13 16.07% 4,165,614 4,582,611 804,417 11,288,850 1.215 -29,314 $282.22 ($0.73) 0.0 0.0
11 55.59 17.82% 4,167,341 4,579,729 806,162 11,461,930 1.233 -27,117 $286.55 ($0.68) 0.0 0.0
12 61.05 19.57% 4,164,211 4,582,685 806,080 11,813,870 1.271 -27,807 $295.35 ($0.70) 0.0 0.0
13 66.52 21.32% 4,166,547 4,580,054 806,142 12,254,340 1.319 -24,522 $306.36 ($0.61) -0.1 0.0
14 71.98 23.07% 4,165,471 4,580,987 807,835 12,469,980 1.342 -20,021 $311.75 ($0.50) -0.1 0.0
15 77.44 24.82% 4,164,902 4,578,882 809,664 12,672,490 1.364 -15,435 $316.81 ($0.39) -0.1 0.0
16 82.91 26.57% 4,165,956 4,578,669 809,595 13,042,160 1.403 -5,416 $326.05 ($0.14) -0.1 0.0
17 88.37 28.32% 4,166,620 4,576,274 812,403 14,012,980 1.508 -2,845 $350.32 ($0.07) -0.1 0.0
18 93.83 30.08% 4,167,561 4,575,069 812,892 14,794,290 1.592 28,966 $369.86 $0.72 -0.1 0.0
19 99.30 31.83% 4,165,090 4,573,596 816,546 15,054,050 1.620 24,198 $376.35 $0.60 -0.1 0.0
20 104.77 33.58% 4,166,216 4,574,133 815,088 15,300,520 1.646 35,610 $382.51 $0.89 -0.1 0.0
21 110.23 35.33% 4,168,606 4,571,452 816,873 15,895,380 1.710 49,996 $397.38 $1.25 -0.1 0.0
22 115.70 37.08% 4,168,846 4,569,413 818,783 17,108,720 1.841 67,031 $427.72 $1.68 -0.1 0.0
23 121.16 38.83% 4,169,679 4,568,267 819,559 17,843,230 1.920 66,711 $446.08 $1.67 -0.1 0.0
24 126.63 40.59% 4,167,781 4,569,169 820,234 18,060,250 1.943 85,153 $451.51 $2.13 -0.1 0.0
25 132.09 42.34% 4,169,917 4,568,677 820,980 18,398,330 1.980 113,207 $459.96 $2.83 -0.1 0.0
26 137.56 44.09% 4,170,845 4,568,164 823,318 19,647,830 2.114 118,675 $491.20 $2.97 -0.1 0.0
27 143.03 45.84% 4,175,218 4,563,483 825,749 21,822,100 2.348 171,073 $545.55 $4.28 -0.1 0.1
28 148.50 47.60% 4,171,833 4,560,732 830,305 22,875,490 2.462 180,631 $571.89 $4.52 -0.1 0.0
29 153.97 49.35% 4,171,689 4,562,919 829,314 23,150,140 2.491 198,766 $578.75 $4.97 -0.1 0.0
30 159.43 51.10% 4,171,731 4,565,792 828,805 23,721,600 2.553 205,712 $593.04 $5.14 -0.1 0.0
31 164.90 52.85% 4,175,640 4,563,682 829,333 24,454,090 2.631 269,371 $611.35 $6.73 -0.1 0.0
32 170.37 54.61% 4,174,019 4,562,495 832,865 25,273,610 2.720 279,533 $631.84 $6.99 -0.1 0.0
33 175.84 56.36% 4,177,251 4,560,723 833,978 26,621,130 2.865 288,026 $665.53 $7.20 -0.1 0.0
34 181.31 58.11% 4,176,817 4,562,460 834,863 27,813,710 2.993 312,594 $695.34 $7.81 -0.1 0.0
35 186.78 59.87% 4,176,635 4,561,589 838,714 29,016,580 3.122 348,636 $725.41 $8.72 -0.1 0.0
36 192.25 61.62% 4,182,555 4,556,475 840,400 30,399,960 3.271 412,931 $760.00 $10.32 -0.1 0.0
37 197.72 63.37% 4,184,189 4,557,148 842,698 31,921,580 3.435 434,304 $798.04 $10.86 -0.1 0.0
38 203.20 65.13% 4,182,709 4,556,475 846,548 33,963,500 3.655 489,678 $849.09 $12.24 -0.1 0.0
39 208.67 66.88% 4,185,794 4,553,933 847,759 35,386,150 3.808 553,282 $884.65 $13.83 -0.2 0.0
40 214.11 68.63% 4,184,275 4,546,483 846,955 36,247,870 3.906 646,652 $906.20 $16.17 -0.2 0.0
41 219.40 70.32% 4,135,873 4,494,894 837,315 36,279,320 3.956 664,013 $906.98 $16.60 -0.2 -0.1
42 224.09 71.82% 3,852,087 4,190,831 778,776 32,837,310 3.843 593,913 $820.93 $14.85 -0.5 -0.2
43 227.58 72.94% 2,922,475 3,195,238 590,124 22,538,390 3.466 330,975 $563.46 $8.27 -1.3 -0.7
44 229.75 73.64% 1,426,954 1,571,225 285,954 8,961,494 2.811 89,687 $224.04 $2.24 -2.6 -1.5
45 230.98 74.03% 351,051 391,620 70,555 1,651,729 2.090 -1,962 $41.29 ($0.05) -4.5 -2.7
46 231.70 74.26% 36,733 41,043 7,322 127,911 1.544 -1,003 $3.20 ($0.03) -6.8 -4.3
47 232.14 74.40% 1,432 1,550 292 4,082 1.282 -172 $0.10 $0.00 -9.5 -6.0
48 232.31 74.46% 21 24 6 59 1.204 -5 $0.00 $0.00 -13.0 -8.4

Don Schlesinger · 01-12-2009, 09:35 PM

> Conversely, RSA would probably increase this number by
> a bit.

Not enough to matter at all.

> The conventional number is 2.7 cards used per player,
> but I believe this assumes more than 1 player at the
> table.

For just the players' hands, it makes no difference how many players there are. But, if you count the dealer's hand in the average, as opposed to stating a different value for his hand alone, then, clearly, it will be lower with a single player, for the reason you state below.

> But when heads up, the dealer does not have to
> take additional hit cards when the player busts, which
> leads me to believe the avg should be smaller than 2.7
> cards per player.

Again, it isn't lower per PLAYER, because the dealer isn't a player; he's the dealer! :-) But, having the dealer play against a lone player lowers the overall average slightly, because the player breaks one hand in six, at which point the dealer doesn't complete his hand. So, instead of the dealer's average being six hands of 2.7, it is now five hands of 2.7 and one hand of 2.0, making the average more like 2.6.

Note that, as more players are added, making the likelihood that all will break smaller and smaller, the 2.6 gravitates closer and closer to 2.7. Finally, since the dealer operates under different rules to play his hand than does the player, even if the dealer completes his hand every round, I think that his average is slightly lower than 2.7, to begin with -- maybe 2.65, but I'm not certain about that.

> The rules pertaining to my game are 6D, S17, DAS, DA2,
> heads up.

Well, the heads-up definitely matters, as per above.

Don

MJ · 01-12-2009, 11:54 PM

> I simmed your game twice in CVData: the first time
> with LS, and the second time without it. Both sets of
> output are shown below.

I appreciate it.

> If we confine our observations to only the first 38
> rounds of each shoe (after that point the cut card
> starts to have a significant impact), we can see on
> average the LS game uses 2.711 cards per hand per
> round, while the no LS game uses 2.733 cards per hand
> per round: a difference of 0.022 cards per hand per
> round.

Can you explain how you arrive at these figures? Thanks for any further assistance.

MJ

Dog Hand · 01-14-2009, 07:54 PM

MJ,

If you look back at the LS output, you'll see the average number of cards in the discard rack at the start of each round are as follows: #1, 1.00; #2, 6.41; #3, 11.81...

This means that, on average Round 1 uses (6.41-1.00)/2 = 2.705 cards; Round 2 uses (11.81-6.41)/2 = 2.700, etc.

I used Excel to calculate the value for each of the first 38 rounds, then average all 38 values to get the reported results of 2.711 with LS, and 2.733 w/o LS.

Hope this helps!

Dog Hand

Aruuba · 01-14-2009, 08:02 PM

> Can you explain how you arrive at these figures?
> Thanks for any further assistance.
> MJ

I'm guessing he subtracted the burn card from the total cards used after 39 rounds and divided by 38 rounds played?

I guess this 2.7 avg is for a BS player with no cut or burn card?

Why, if Dog's sim was for only 1 player heads up, is it not closer to the 2.6 that Don mentioned because the dealer wouldn't complete player busted hands?

How significant, if much at all, would the cut-card be in using this avg of 2.7?

Would it change much for a counter using indexes?

Just wondering if it would matter much over time if a counter used pen level (without taking into account any cut-card effect ie 234 cards in this case)and number of players at table for a shoe to later estimate rounds played. Maybe rather doing it that way than just using time played to estimate rounds played?

Thread: MJ: Cards used per player per hand

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