# Blackjack Combinatorial Analysis by Simulation

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One of the most valuable resources for researching blackjack is appendix 4 at the Wizard of Odds website.  In that appendix, Mike Shackleford gives the combinatorial analysis (CA) for the version of six-deck blackjack most commonly played by high-rollers. I used Shackleford’s work in my analysis of Don Johnson’s method for beating loss rebates (see this post). Recently, a reader of this blog asked if I knew a program that gave the CA for other rule sets. I asked Mike how he did his CA and he told me that his results in Appendix 4 were obtained by simulation. In other words, he ran a bunch of rounds, recorded the number of times each event occurred, imported that data into a spread sheet, and completed his analysis from there. In this post, I present CA by simulation for six other blackjack rule sets.

In what follows, I use the following abbreviations:
• 2D (two decks), 6D (six decks).
• DOA (double on anything), player can double on any first two cards.
• DAS (double after split), player can double after splitting.
• H17, dealer must hit soft-17.
• S17, dealer must stand on soft-17.
• LSR (late surrender), player can surrender his initial two cards after the dealer checks for blackjack.

The results I present below were obtained by simulations of four billion (4,000,000,000) hands of blackjack each.

### Two Decks

The following tables give the CA by simulation for the following rule sets:
• 2D, H17, DOA, DAS
• 2D, S17, DOA, DAS
• 2D, S17, DOA, DAS, LSR

The following table summarizes the two-deck results:

### Six Decks The following tables give the CA by simulation for the following rule sets:

• 6D, H17, DOA, DAS
• 6D, S17, DOA, DAS
• 6D, S17, DOA, DAS, LSR

The following table summarizes the six-deck results: