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# Gaming Guru

### Playing the Soft 18

7 April 2002

One of the hands that is frequently misplayed in the game of Blackjack is the soft 18 (such as Ace-Seven).  In fact I have had dealers go right by me when I held this hand, assuming I was going to stand and not waiting for a hand signal from me.  It turns out that one should hit this hand against a dealer's up card of 9 or 10.  For multiple decks one should also hit this hand against the Ace, although in one and two decks this rule has to be modified depending upon whether or not the dealer stands on the soft 17.

It is not hard to see why some players are reluctant to take this advice.  It would not be uncommon to hit this hand into a stiff and then bust the stiff drawing again.  I know the feeling.  I have similar feelings when I hit the hard 12 against the dealer's 2 and 3 but I do it.  In this article I am going to take a look at the question of hitting or standing with the soft 18 when the dealer's up card is a 9.  By doing the calculations we can get a feel for the dynamics of the play and see why hitting is the right play.

Blackjack calculations are a humbling experience.  Early on in the 60s and 70s, the strategy decisions were made by doing computer simulations; the name of the recently late Julian Braun is associated with these simulations.  It was not until 1979 and the publication of his book Theory of Blackjack (Huntington Press, Las Vegas) that the late Peter Griffin showed that exact calculations could be carried out, although not without monumental effort.  There is, however, an approximate calculation that one can make that works well for multiple decks.  It is to assume an infinite deck.  Peter discussed this on pages 170 and 171 of his book.  Now, an infinite deck isn't really infinite; it is just a single deck in which every card drawn is replaced with another card of the same rank.  This means, for example, that if a 7 is drawn from the deck, then on the next draw the probability of drawing a 7 is the same as it was before.  The probabilities for an infinite deck are therefore, p(k) = 1/13 for k = 1 through 9 and p(10) = 4/13.  These are the assumptions we will make in our calculations.

The first thing we are going to do is figure out what the dealer's probabilities are (notice that if we weren't using an infinite deck we would have to know the player's draw before making this calculation).  It will be convenient to separate the hands into hard hands and soft hands as follows.  Hard hands will be hands that do not contain an Ace and will range from 9 to 26.  Soft hands will be any hand containing an Ace where the first Ace counts as 11 and the rest count as 1.  Soft hands, therefore, range from 20 (Ace-Nine) to 36.  Let me explain.  Using my definitions, soft hands 22 through 26 correspond to the usual hard hands 12 through 16 that are converted soft hands, soft hands 27 through 31 correspond to final dealer hard totals of 17 through 21, and hands 32 to 36 are dealer busts of 22 through 36.  Here goes.

The symbol h(k) will represent the probability of a hard total of k.  Clearly h(9) = 1 before the hole card is exposed.  Now h(10) = 0 because if an Ace is drawn it is not a hard hand, it is a soft 20, which we'll deal with later.  h(11) is 1/13, which is just the probability of drawing a 2 to the 9.  h(12) is also 1/13 since we can draw a 3 to the 9 but cannot draw a two to the hard 10 because there isn't a hard 10.  What about the hard 13?  Well we can draw 4 to the 9 or 2 to the 11 (3 doesn't work does it?).  In symbols

 h(13) = p(2)h(11) + p(3)h(10) + p(4)h(9) (1)
 h(13) = (1/13)(1/13) + (1/13)0 + (1/13)1 (2)

This number is 0.082840237.  I'll bet you could do h(14) now.  You see this is how we step our way up the probability tree.  Here are the final results tabulated.

 Hard Total Probability Hard Total Probability 9 1.00000000 18 0.11711775 10 0.00000000 19 0.34788698 11 0.07692308 20 0.04019467 12 0.07692308 21 0.05794615 13 0.08284024 22 0.05202899 14 0.08875740 23 0.04747733 15 0.09512972 24 0.04247050 16 0.10195721 25 0.03711355 17 0.10927489 26 0.03137145
Table 1
Dealer's Hard Probabilities with 9 Up

So, for example, if I wanted to calculate the probability of a hard 19 assuming all of the lower ranked probabilities had already been calculated the calculation in symbols would be:

 h(19) = p(3)h(16) + p(4)h(15) + p(5)h(14) + ... + p(9)h(10) + p(10)h(9) (3)

Thus if you add the probabilities for totals 10 through 16 plus 4 times the entry in 9 and divide by 13 you'll get the number shown in 19.  Try it.

Now for soft totals.  The smallest soft total starting with 9 is 20.  A soft 21 is impossible since the dealer must stand on the soft 20 and hitting a hard 10 with an Ace is impossible since there is no hard 10.  A soft 22 (converted hard 12) can result from hitting a hard 11 with an Ace.  A soft 23 can result from hitting the hard 12 with an Ace or the soft 22 with an Ace (now you can see why I defined 'hard' and 'soft' the way I did).  A soft 24 can result from hitting a hard 13 with an Ace, a soft 22 with a 2, or a soft 23 with an Ace.  And so on.  Here is the table.

 Soft Total Probability Soft Total Probability 20 0.07692308 29 0.00287769 21 0.00000000 30 0.00287769 22 0.00591716 31 0.00287769 23 0.00637233 32 0.00424319 24 0.00731767 33 0.00389306 25 0.00833573 34 0.00362104 26 0.00946712 35 0.00329308 27 0.01072056 36 0.00291296 28 0.00287769
Table 2
Dealer Soft Probabilities with 9 Up

For example, if one wanted to calculate the probability of a soft 26 (converted hard 16) the calculation in symbols would be

 s(26) = p(1)h(15) + p(1)s(25) + p(2) s(24) + p(3)s(23) + p(4)s(22) (4)

In other words, if we add the entries 22 through 25 in Table 2 and then add the entry from total 15 in Table 1 and divide the result by 13, we get the entry in space 26 in Table 2.  Try it.

With the above numbers in hand we can now calculate the dealer's final probabilities.  For example, to get the probability of the dealer's final hand being a 20, we would add the entry in Table 1 for h(20), the entry in Table 2 for s(20), and the entry in Table 2 for s(30) (a converted hard 20).  Here is the result:

0.04019467
0.07692308
0.00287769
0.11999544

Easy!  When we calculate the totals for 22 through 26 we can add these together to represent the probability of a dealer bust.  Here are the results.

 Final Total Probability 17 0.11999544 18 0.11999544 19 0.35076467 20 0.11999544 21 0.06082384 Bust 0.22842517 Total - 1.00000000
Table 3
Dealer's Final Probabilities with 9 Up

With Table 3 in hand we can determine the player's expected return by standing on a total of 18 (soft or hard) when the dealer has a 9 up.  Here it is:

 Dealer Total Probability Payoff Product 17 0.11999544 +1 +0.11999544 18 0.11999544 0 +0.00000000 19 0.35076467 -1 - 0.35076467 20 0.11999544 -1 - 0.11999544 21 0.06082384 -1 - 0.06082384 Bust 0.22842517 +1 +0.22842517 Total - - 0.18316334
Table 4
Expectation for Player Standing on 18

So there you have it; standing on the 18 against the dealer's 9 will result in an expected loss of a bit over 18 cents.  We are going to need the standing expectations for the other totals, calculated the same way of course, and here they are.

 Stand Total Expectation 17 - 0.42315422 18 - 0.18316334 19 +0.28759677 20 +0.75835688 21 +0.93917616
Table 5
Player's Standing Expectations with 9 Up

What if the player hits?  Here we have to suppose that the hard hit/stand strategy has already been determined, that is, against a 9 the player will hit hard 16 and below and stand on 17 or more.  In actual practice these determinations have to be handled before addressing the soft hands.  To determine player soft strategy we start working from the top down.  Not surprisingly the results are that the player should stand on the soft 19, 20 and 21 against the 9 and we will assume that these determinations have been made using an analysis like the one that follows.  Recalling how I defined 'soft' earlier, starting with a soft 18, hitting can only produce more soft hands.  Thus our next job is to obtain a table similar to Table 2 except this time it is for the player.  Here it is:

 Soft Total Probability Soft Total Probability 19 0.07692308 28 0.34219357 20 0.07692308 29 0.03450126 21 0.07692308 30 0.03450126 22 0.07692308 31 0.03450126 23 0.08284024 32 0.05225274 24 0.08921256 33 0.04770108 25 0.09607507 34 0.04279929 26 0.10346546 35 0.03752044 27 0.11142434 36 0.03183553
Table 6
Soft Total Probabilities Hitting Soft 18

Note that the only way to obtain 19, 20, or 21 is to hit the soft 18 with an Ace, two or three, respectively.  The only way to get a soft 22 is to hit the soft 18 with a four since there are no hard hands and we don't hit the soft 19, 20 , nor 21.  Thus the first four entries in Table 6 are all 1/13 or 0.07692308.  The rest of the table is filled in just as Table 2 was done.  For example, the soft 34 (hard 24) is calculated as

 s(34) = p(10)s(24) + p(9)s(25) + p(8)s(26) (5)

Note that the pattern stops at 26 because 27 through 31 (hard 17 through 21) are not hit and 32 and 33 are already busted hands.

With Table 6 in hand we can now calculate the player probabilities.  For example, the probability that the player ends up with a 19 is s(19) + s(29).  In general, if k is 18 through 21 the probability is s(k) + s(k + 10) and for k equal to 17 or 18 or k > 21 the probability is just s(k + 10).  As with the dealer, adding the probabilities for the hard 22 through 26 we have the probability of a bust.  Here are the results.

 Player Total Probability 17 0.11142434 18 0.34219357 19 0.11142434 20 0.11142434 21 0.11142434 Bust 0.21210907 Total - 1.00000000
Table 7
Player Probabilities by Hitting Soft 18

Now we have everything we need to finish up.  Combining the information in Table 7 with that in Table 9 we can calculate the player's expected return by hitting the soft 18 against the dealer's 9.  When we hit the soft 18 and play according to the hard hit/stand strategy we get one of the totals in Table 7 and then we stand.  The expected return can be calculated using the following table.

 Player Total Probability Expectation Product 17 0.11142434 - 0.42315422 - 0.04714968 18 0.34219357 - 0.18316334 - 0.06215378 19 0.11142434 +0.28759677 +0.03204528 20 0.11142434 +0.75835688 +0.08449941 21 0.11142434 +0.93917616 +0.10464708 Bust 0.21210907 - 1.00000000 - 0.21210907 Total - - 0.10022076
Table 18
Expected Return Hitting Soft 18 vs. 9

At last we have it.  The player's overall expected return by hitting the soft 18 is around a dime loss.  Earlier we discovered that by standing on the soft 18 the player's expected loss was around 18.3 cents.  The conclusion is inescapable; the player should hit the soft 18 against the 9.  It follows that the player should also hit the soft 18 against the 10 since a 10 up has stronger dealer potential than a 9.  The Ace is a different matter and requires a separate analysis -- another day.

Notice that when facing a soft 18 against the 9 you are in an overall losing situation no matter what you do.  I have noticed that when you tell players that by doing so and so you'll win more overall, they readily buy it.  When you tell them to do so and so and you'll lose less it seems to be a much harder pill to swallow.  That is the case here.  The fact is that an 18 just isn't a very good hand and when it's soft you have the chance to do something about it.

If there is any disadvantage to hitting the soft 18 against the 9 and 10 it is this.  If you hit into a stiff and then bust there is a chance there will be a ploppie (see Frank Scoblete for a definition) sitting at your table who will start complaining that you just screwed up a good hand and blah, blah, blah.  Your reply?  Tell him (or her, they come in both sexes) to read this article and they'll recognize themselves.  Naw, come to think of it, they probably couldn't read this article.  See you next month.

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Best of Donald Catlin
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