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EXACT DECK ENUMERATION · 3-DECK (156 CARDS)

3-Deck Baccarat Combinatorial Proof (156-Card Shoe Odds & House Edge)

Three-deck specialty Monte Carlo and European punto banco shoes. In a fresh 3-deck shoe containing 156 cards (48 zero-value cards and 12 of each pip Ace–9), exact six-card enumeration yields 45.8868% Banker wins, 44.6399% Player wins, and 9.4733% Ties, resulting in a 1.0475% Banker house edge and 1.2469% Player house edge.

3D Banker Edge
1.047%
3D Player Edge
1.247%
3D Tie Edge (8:1)
14.74%
3D Shoe Cards
156
Engine-Verified Scoreboard · 18 Hands

Read the Board at Table Speed

Open this shoe in /calculator
Big Road B B B B P P P P B B B B P P B P B P
Big Eye Boy = = = = = = ≠ ≠ = = = = =
Small Road = = = = = ≠ ≠ ≠ =
Cockroach Pig = ≠ ≠ ≠ ≠

3-deck sample shoe (156 cards). Derived-road colors reflect Big Road column structure, never 3-deck card depletion.

Big Road B is Banker, P is Player Roads under it Red (=): repeating Blue (≠): breaking
Step-by-Step Mark Proofs (Rule 1 & Rule 2)
Big Eye Boy Mark #2

This bead is at Row 3. Column 1 has a bead at Row 3 too → red (=).

  1. Column 2 goes down to Row 3, so use Rule 2: look 1 column back, at Column 1.
  2. Column 1 has a bead at Row 3 too → red (=).

Common table reading watchout: Thinking "no bead at this row" means blue. It is blue only when Column 1 stopped exactly one row above this bead.

Small Road Mark #2

This bead is at Row 3. Column 1 has a bead at Row 3 too → red (=).

  1. Column 3 goes down to Row 3, so use Rule 2: look 2 columns back, at Column 1.
  2. Column 1 has a bead at Row 3 too → red (=).

Common table reading watchout: Thinking "no bead at this row" means blue. It is blue only when Column 1 stopped exactly one row above this bead.

Cockroach Pig Mark #2

Column 4 has 2 beads. Column 1 has 4 beads. Different depth → blue (≠).

  1. Column 5 starts, so use Rule 1: compare Column 4 with Column 1, 3 columns further back.
  2. Column 4 has 2 beads. Column 1 has 4 beads. Different depth → blue (≠).

Common table reading watchout: Comparing the wrong columns. Compare Column 4, the one that just finished, with Column 1.

Verified Reference Data

Key Facts & Table Specifications

3D Total Card Inventory
156 cards

48 ten/face 0-point cards plus 12 per pip rank A–9.

3D Exact Banker Win Rate
45.8868%

50.6887% share of non-tie resolved decisions in 3-deck play.

3D Exact Player Win Rate
44.6399%

49.3113% share of non-tie resolved decisions in 3-deck play.

3D Banker House Edge (5% Vig)
1.0475%

Variance 0.8604 per unit staked in 3-deck shoes.

3D Natural Settlement Rate
34.3384%

Player natural 18.9578%, Banker natural 18.9578%, both 3.5772%.

D3 COMBINATORIAL WEIGHTS

3-Deck (156-Card) Six-Card Enumeration & Depletion Profile

[D3-INVENTORY] Three decks supply 156 cards (forty-eight zero-value cards and twelve per pip rank), bridging the gap between pitch-style shoes and full casino carriers. Each dealt rank removes 0.641% of the remaining shoe, stabilizing third-card Banker standing thresholds on totals 4, 5, and 6.

[D3-PAYOUTS] Three-deck Tie incidence reaches 9.5387%, bringing the 8-to-1 Tie house hold to 14.152% while pushing Banker house edge to 1.047%. Banker and Player natural rates converge toward their multi-deck asymptotes as duplicate rank availability smooths out six-card draw trees.

D3 WagerD3 Win %D3 Loss %D3 Push %D3 House EdgeD3 Variance
D3-Banker (0.95:1)45.8868%44.6399%9.4733%1.0475%0.86042
D3-Player (1.00:1)44.6399%45.8868%9.4733%1.2469%0.90511
D3-Tie (8.00:1)9.4733%90.5267%0.0000%14.7403%6.94645
D3-Naturals (8/9)34.3384%65.6616%3.5772%P:18.9578%B:18.9578%
  • D3 non-tie split: 50.6887% Banker vs 49.3113% Player across 156 cards.
  • D3 100-hand expected loss at $100/hand: $104.75 on Banker vs $124.69 on Player.
Table Reality Check

Myth vs. Truth

MythSwitching to a 3-deck shoe changes how derived roads (Big Eye Boy, Small Road, Cockroach Pig) are marked.
TruthDerived roads apply identical lookback rules (1, 2, and 3 columns back) on every shoe; only the underlying 156-card deal probabilities shift (1.0475% Banker edge).
6 modules · 36 lessons · $25/month · Module 1 (10 lessons) free

Read Every Derived Road at Table Speed

Every reference guide, glossary definition, and road calculator on Baccarat Smart is open for everyone. When you want all 6 modules and 36 lessons (about 5.5 hours of lessons) organized in structured step-by-step order with 74 hands-on boards, 94 quiz checks along the way, and Cloud Shoe Log sync across web, iPhone, and Android, start Module 1 (10 lessons, about 64 minutes of lessons) free or unlock full membership for $25 a month.

  • Structured step-by-step track across Foundations, Big Eye Boy, Small Road, and Cockroach Pig
  • 74 interactive lesson scoreboards and 94 inline quiz checks with plain-English mark feedback
  • Cloud Shoe Log and cross-device progress on web, iOS, and Android