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# Playing the Soft Eighteen

4 September 2004

One of the frequently misplayed Blackjack hands is the soft eighteen, that is, a hand comprised of an Ace and a count of seven. Clearly if the dealer shows a bust card, that is, a two through six, then the player is in a favorable situation if he just stands with the eighteen. Clear also is the situation wherein the dealer shows a seven or eight. Since the dealer's most likely hand is a seventeen or eighteen, standing on the soft eighteen makes sense. What about the dealer's nine, ten, or Ace?

The Ace is a bit complicated and I'll discuss that later. The correct play against the nine and ten is to hit the soft eighteen. Why? Well the precise answer is that the player's expected return by hitting is higher than it is by standing with this hand. I'm sure that you can find the details of that in the many Blackjack books that are on the market. My purpose in this article is to just give you a plausible argument that hitting is the right thing to do so that you'll be comfortable making the right play.

More than once I have had dealers go by me in this situation and when I called them on it they apologized and told me that most players don't hit the soft eighteen at any time. So, I figure a little pep talk is appropriate.

Let us look at the situation where the dealer's up card is a ten, the play is head to head, and the hand is the first hand dealt. In this scenario the player (you) holds Ace-Seven. If the game is a double-deck game and the dealer doesn't have Blackjack, then there are 31 tens left in the deck and 8 nines or 39 cards that will beat you immediately; the probability of that happening is 39/101 or 0.3861. The 8 eights push and the rest of the hands (assuming the dealer hits soft seventeen) will sometimes win and sometimes lose. For example, if the dealer has a four in the hole, a probability of 0.0792, then there are 23 cards out of the remaining 100 that will beat you, a probability of 0.23 (remember, one of the sevens is in your hand). Together we have a probability 0.0792 x 0.23 or about 0.018 that the dealer will beat your eighteen on the first draw. Of course, if the dealer draws an Ace or a two he still might beat you, so the actual figure that the dealer will beat you with a four in the hole is somewhere around 0.02. If the dealer shows a five then there are 24 cards that will beat you and this gives us a probability of around 0.019 of being beaten on the dealer's first draw. Again, the dealer still might win if he draws an Ace so again I'll estimate the probability of losing when the dealer has a five in the hole at around 0.02. The same argument will hold for two, three, six and seven, though for the lower cards there are more chances for the dealer to beat you so the 0.02 figure is probably low. If we add the 0.02 figure 6 times to 0.3861 we end up with 0.5061, a better than even chance that you'll lose in this situation if you stand (the actual figure is a bit smaller than this but is still greater than 0.5).

So what if you draw? Well, there are 23 cards that will improve your hand and 32 cards in the deck that will leave it the same. Of course, even if you hit into a stiff you still might improve your hand or leave it the same. So we can say with complete confidence that over half of the remaining draws will either improve your hand or leave it the same.

Here, then, is the situation. If you stand you have a better than even chance that you will lose and if you hit you have a better than even chance that your hand will be at least as good as it is now. This is not positive proof that hitting is the proper play but it certainly makes it seem like the reasonable thing to do. As I said earlier, the correct analysis requires a computer and involves calculating the expected returns by standing or hitting and seeing which one is larger.

In some single-deck situations one should stand with soft eighteen versus the dealer's Ace, in particular, if the soft seventeen rule is not in effect. For multiple decks, however, the rule of thumb is that one should always hit the soft seventeen versus the dealer's nine, ten, and Ace. I hope the above plausibility argument helps to make you comfortable with this decision. I should also point out that if your soft eighteen is an Ace-Seven then you should double the hand versus the dealer's Three, Four, Five, and Six.

Next month I am going to (almost) forgo the mathematics and tell you a story about a strange road in California. Just to keep our hand in, though, here is a brain teaser for you; I'll tell you the answer next month.

In a 52-card deck, 13 of the cards have been placed in the deck upside down (all of the clubs, for example). Your problem is to take the deck, put it behind your back, and split it into two packets of cards such that when you place both packets on a table each has the same number of cards facing up. Good Luck!

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers