Mathematical Analysis
Of The Following Side Bet For
Card Games
May 20, 2003
| INSTALLATIONS (Land) |
|
Of The Following Side Bet For
Card Games
Prepared By
|
RULES
Following are the rules for
Blackjack Block Bonus™
Blackjack Block Bonus™ is a side bet that can be played on any blackjack game.
The game is based on the initial two cards and the dealer’s up card.
1. The side bet (Blackjack Block Bonus™) can be played on any blackjack game. There is no strategy to the side bet (Blackjack Block Bonus™) so it will not interfere with the blackjack play.
2. The side bet (Blackjack Block Bonus™) shall win or lose depending only on the player’s initial 2 cards and the dealer’s up card.
3. If the player’s cards are different rank and suit, and 1 of them matches the dealer’s suit and exceeds the dealer’s rank the player shall have a “Normal Block” which pays 2 to 1.
4. If the suit of both player cards match the suit of the dealer card, and at least 1 of the player's card exceeds the dealer card in rank, then the player shall have a “Flush Block”, which pays 5 to 1.
5. If the player has an unsuited pair, and 1 of them match the dealer card in suit, and exceed the dealer card in rank, then the player shall have a “Pair Block”, which pays 10 to 1 with 2 or more decks, and 15 to 1 in a single deck game.
6. If the player’s 2 cards are same in both rank and suit, and that suit matches the dealer’s suit, and the player’s rank exceeds the dealer’s rank, then the player shall have a “Ultimate Block” which pays from (35 to 60) to 1, depending on the number of decks. However this win does not apply to a single deck game since it is impossible to have two identical cards in a single deck.
7. If at least 1 of the player cards matches the dealer’s card in both rank and suit, and other card (if any) does not match the dealer’s suit or is less in rank than the dealer’s rank. Then the player shall have a tie, which is a push.
Table 1 shows some examples.
|
Table 1: some examples |
||
|
Player cards |
Dealer cards |
Outcome (Player) |
|
Js, Qc |
5d |
Loss |
|
Js, Qs |
5d |
Loss |
|
Js, Qc |
Ks |
Loss |
|
Js, Qs |
Ks |
Loss |
|
Js, Js |
Ks |
Loss |
|
Js, Qs |
Qs |
Push |
|
Js, Qc |
Qc |
Push |
|
Js, Js |
Js |
Push |
|
Js, Qc |
5c |
Win (Normal Block) |
|
Js, Qs |
Js |
Win (Flush Block) |
|
Js, Ks |
Qs |
Win (Flush Block) |
|
Js, Jc |
5s |
Win (Pair Block) |
|
Js, Js |
5s |
Win (Ultimate Block) |
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ANALYSIS
Table 2 shows the number of combinations of each outcome, according to the number of decks.
|
Table 2: The number of combinations of each outcome, according to the number
of decks |
|||||||
|
Decks |
Loss |
Push |
Normal Block |
Flush Block |
Pair |
Ultimate Block |
Total |
|
1 |
103688 |
0 |
22464 |
4576 |
1872 |
0 |
132600 |
|
2 |
838864 |
18720 |
179712 |
39104 |
14976 |
1248 |
1092624 |
|
3 |
2841696 |
84552 |
606528 |
134784 |
50544 |
5616 |
3723720 |
|
4 |
6748352 |
225888 |
1437696 |
322816 |
119808 |
14976 |
8869536 |
|
5 |
13195000 |
471120 |
2808000 |
634400 |
234000 |
31200 |
17373720 |
|
6 |
22817808 |
848640 |
4852224 |
1100736 |
404352 |
56160 |
30079920 |
|
7 |
36252944 |
1386840 |
7705152 |
1753024 |
642096 |
91728 |
47831784 |
|
8 |
54136576 |
2114112 |
11501568 |
2622464 |
958464 |
139776 |
71472960 |
Table 3 shows the probability of each outcome, according
to the number of decks.
|
Table 3: The
probability of each outcome, according to the number of decks |
|||||||
|
Decks |
Loss |
Push |
Normal Block |
Flush Block |
Pair |
Ultimate Block |
Total |
|
1 |
0.781961 |
0.000000 |
0.169412 |
0.034510 |
0.014118 |
0.000000 |
1.000000 |
|
2 |
0.767752 |
0.017133 |
0.164477 |
0.035789 |
0.013706 |
0.001142 |
1.000000 |
|
3 |
0.763134 |
0.022706 |
0.162882 |
0.036196 |
0.013574 |
0.001508 |
1.000000 |
|
4 |
0.760846 |
0.025468 |
0.162094 |
0.036396 |
0.013508 |
0.001688 |
1.000000 |
|
5 |
0.759480 |
0.027117 |
0.161623 |
0.036515 |
0.013469 |
0.001796 |
1.000000 |
|
6 |
0.758573 |
0.028213 |
0.161311 |
0.036594 |
0.013443 |
0.001867 |
1.000000 |
|
7 |
0.757926 |
0.028994 |
0.161089 |
0.036650 |
0.013424 |
0.001918 |
1.000000 |
|
8 |
0.757441 |
0.029579 |
0.160922 |
0.036692 |
0.013410 |
0.001956 |
1.000000 |
Table 4 shows the contribution to the total
return of each hand in the
8-deck game.
The lower right cell of table 4 shows an expected return of - 4.96%, in other
words a house edge of +4.96%.
|
Table 4: The contribution to the total
return of each hand in the
8-deck game |
||||
|
Event |
Pays |
Combinations |
Probability |
Return |
|
Loss |
-1 to 1 |
54136576 |
0.757441 |
-0.757441 |
|
Push |
0 to 1 |
2114112 |
0.029579 |
0.000000 |
|
Normal Block™ |
2 to 1 |
11501568 |
0.160922 |
0.321844 |
|
Flush Block™ |
5 to 1 |
2622464 |
0.036692 |
0.183458 |
|
Pair Block™ |
10 to 1 |
958464 |
0.013410 |
0.134102 |
|
Ultimate Block™ |
35 to 1 |
139776 |
0.001956 |
0.068448 |
|
Total |
|
71472960 |
1.000000 |
-0.049590 |
Table 5 shows the win probability and house edge for all number of decks from 1
to 8, along with the pay on the Pair Block and
Ultimate
Block, which vary
according to the number of decks.
|
Table 5: The win probability and house edge
for all number of decks from 1 to 8 |
|||||
|
Decks |
Normal Block | Flush Block |
Pair Block |
Ultimate Block |
House Edge |
|
1 |
2 to 1 |
5 to 1 |
15 to 1 |
N/A |
5.88% |
|
2 |
2 to 1 | 5 to 1 |
10 to 1 |
60 to 1 |
5.43% |
|
3 |
2 to 1 |
5 to 1 |
10 to 1 |
50 to 1 |
4.52% |
|
4 |
2 to 1 | 5 to 1 |
10 to 1 |
40 to 1 |
5.21% |
|
5 |
2 to 1 |
5 to 1 |
10 to 1 |
35 to 1 |
5.61% |
|
6 |
2 to 1 | 5 to 1 |
10 to 1 |
35 to 1 |
5.32% |
|
7 |
2 to 1 |
5 to 1 |
10 to 1 |
35 to 1 |
5.11% |
|
8 |
2 to 1 | 5 to 1 |
10 to 1 |
35 to 1 |
4.96% |
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CONCLUSION
This analysis was conducted using both a pure math and combinatorial approach. Therefore the numbers in this report can be considered to be exactly right.
The
vulnerability to card counters was lightly considered.
Without a full vulnerability study I can make no guarantees but an informal look
suggests the game will be safe under normal penetration of 75% or less.
For example a game in which the remaining suits are distributed according to the
distribution {18.18%, 22.73%, 27.27%, 31.82%}, and equally distributed by ranks,
still has a house edge of 0.78% with 286 total cards.
Such a skewed distribution in my
estimation would be very unlikely in a normal game, except very close to the end
of the shoe.
If you or any interested party have any questions about this analysis please contact me by any of the means below.
E-mail: mail@gamingmath.com
Phone: 702-804-0015
Postal mail: 9200 Sienna Vista Dr., Las Vegas, NV 89117
Thank you for choosing me to analyze Blackjack Block Bonus™ and I wish you success with it.
Regards,
Michael
Shackleford, A.S.A.
Registered Design Number: 4001846
