Single-Deck Basics
May 1st, 2005 by brother9The single-deck blackjack game is practically extinct. You may find a few games here and there, with lousy tacked-on rules to draw in people. Maybe the casino will re-shuffle at 1/3 depth, or maybe even re-shuffle after each hand(!). In any case, it’s still possibly an available game, and the basis of all statistical inquiry into gambling.
That being said, the first statisticians who looked into the game saw a great deal of importance in the likelihood of the dealer busting in order to draw wins for the player as opposed to more modern strategies of bet manipulation, doubling down, and merely trying to get a higher score than the dealer. That is, the original game saw more winners when the dealer busted as opposed to the player having a higher score than the dealer.
That lead to a table of likely outcomes given a dealer’s up card. Here’s the table, expressed in percent rounded to the nearest tenth:
| Final: 17 |
18 | 19 | 20 | 21 | “Natural” Blackjack |
Bust | |
| Up card: 2 |
14.2% | 13.5% | 13.1% | 12.4% | 12.0% | n/a | 34.8% |
| 3 | 13.4% | 13.3% | 12.6% | 12.3% | 11.5% | n/a | 37.0% |
| 4 | 13.2% | 11.6% | 12.3% | 11.8% | 11.4% | n/a | 39.7% |
| 5 | 12.1% | 12.5% | 11.8% | 10.5% | 10.8% | n/a | 42.3% |
| 6 | 16.8% | 10.7% | 10.8% | 10.1% | 9.8% | n/a | 41.7% |
| 7 | 37.3% | 13.9% | 7.8% | 7.9% | 7.3% | n/a | 25.8% |
| 8 | 13.1% | 36.3% | 13.0% | 6.8% | 7.0% | n/a | 23.7% |
| 9 | 12.2% | 10.4% | 35.8% | 12.2% | 6.1% | n/a | 23.3% |
| 10 | 11.5% | 11.3% | 11.5% | 32.9% | 3.6% | 7.8% | 21.4% |
| Ace | 12.8% | 13.1% | 13.0% | 13.1% | 5.1% | 31.4% | 11.5% |
For instance, if the dealer is showing a 7, the chances of ending up with a 19 after playing the hand out is 7.8%; the odds of a 6 becoming a 20 is 10.1%. It’s interesting to note that the propabilities of a dealer showing 9 making 17 are exactly the same as the dealer making 20: 12.2256%.
And if you know that the dealer does not have blackjack, the tables for 10 and A change substantially:
| Final: 17 |
18 | 19 | 20 | 21 | “Natural” Blackjack |
Bust | |
| Up card: 10 |
12.5% | 12.3% | 12.5% | 35.7% | 3.9% | n/a | 23.2% |
| Ace | 18.7% | 19.1% | 18.9% | 19.1% | 7.5% | n/a | 16.7% |
Obviously in the normal course of a 6 deck game the odds will be slightly different, but once you know the basis, you can work up from here. For instance, if the dealer shows a 10 and does not have a blackjack, they are now more likely to bust. This is equivalently true in a multiple-deck game.
