(Too Much) More on 16 v T,7: Part 1
I had to break this post up into two parts as it was too long. Clearly I've got too much time on my hands. ;-)
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The Mayor's answer is spot on, of course, but I thought I might add in a few extra details, since I've noticed lots of players intuitively or instinctively seem to feel that standing on 16 v 7 makes more sense than standing on 16 v T, given the strength of the dealer's T up.
From the outset it should be acknowledged that the player's fear of the dealer's T up is well placed. You are in a worse position holding 16 v T than 16 v 7. Griffin's single-deck figures give basic-strategy expectations of -.540 and -.415, respectively (TTOBJ, 6th ed., p.121).
However, the choice over whether to hit or stand is a separate issue.
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We can get some insight into the matter by considering the effects of removing individual cards for each decision. From Griffin, pp.74-85 we have the following effects of removal (EORs), listed from ace to ten:
16 v T: -0.49, -0.29, -0.80, -1.73, -2.57, 1.65, -0.71, -0.06, 0.55, 1.12; m = -0.45, SSr = 19.1
16 v 7: -1.88, -1.93, -2.44, -2.78, -2.33, 1.80, 2.10, 2.32, 2.77, 0.59; m = 6.07, SSr 48.2
where m is the full-deck favorability of hitting the hand rather than standing and SSr is the sum of squares of the effects of removal.
The above entries deserve a quick explanation:
a) The first ten entries for each hand tell you how the favorability of hitting is altered by removing each card. For instance, removing one 6 from the deck increases the favorability of hitting 16 v T by 1.65%. Similarly, removing one 4 decreases the favorability of hitting 16 v 7 by 2.78%.
b) The value for m tells us how much better it is to hit than stand. Hitting 16 v 7 gives an expectation that is 6.07% better than standing. If we remove the dealer's upcard, this is modified to 6.07 + 2.10 = 8.17% better to hit than stand. The situation is much closer for 16 v T. In fact, if our cards were to be drawn from a full 52-card deck, it would actually be .45% better to stand. However, removing the dealer's upcard, we get -.45 + 1.12 = .67% better to hit.
c) The values for the SSr indicate the degree of 'volatility' inherent in the situation. Volatility refers to how rapidly the favorability of hitting changes relative to standing as the deck is depleted. (In other situations, it could relate to the choice between hitting and splitting, standing and splitting or hitting and doubling.) A high SSr simply reflects the presence of at least some EORs with large absolute values, which implies that removing some cards should ideally have a big effect on our decision.
Looking at the EORs, we can see that 16 v T is a much closer decision off the top of the deck. On the other hand, 16 v 7 is more volatile, so it will still be possible for standing to be correct sometimes.
So we get back to the original question: Why is 16 v T a closer call than 16 v 7? A further question will suggest itself once we have dealt with the first one: How well will our count systems detect variations in the favorability of hitting versus standing?
As will be seen, our reticence to stand on 16 v 7 is partly due to the large full-deck favorability of hitting, but also partly because of the ineptness of standard systems in detecting situations where it would be beneficial to stand.
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If you look at the EORs for 16 v T you will notice a few things:
a) The EORs for aces, 2s and 3s are small.
b) The EORs for 7s, 8s and 9s are small.
c) The EORs for tens are moderate.
d) The key cards are the 5s and 6s.
The reason for a) is that removing aces, 2s or 3s has contradictory effects on the favorability of hitting 16 v T. The primary effect of removing a 2 or 3 is to make hitting less palatable, since we are more likely to bust. However, working against this is the fact that removing a 2 or 3 makes it less likely that the dealer will bust (from a 2 or 3 in the hole), partially dampening any enthusiasm for standing. This impact on the dealer's bust probability is itself dampened though, by the fact that a 2 or 3 will sometimes prevent a dealer bust (eg, when the holecard is a 6). The ace is slightly different. Assuming no ace in the hole (otherwise it wouldn't matter what action we took), the effects of removing an ace on the dealer bust probability is restricted to what happens when the dealer is required to draw a third or later card. Here, sometimes the appearance of an ace causes the dealer to bust (eg, T, 5, A, 6), but sometimes helps the dealer make a hand (eg, T, 6, A or T, 2, A, 4). Overall it turns out that removing an ace marginally increases the dealer's chance of busting off a ten. In terms of the player's hitting prognosis, removing an ace worsens the situation, but only moderately, since an ace is less help than a 2 or 3 anyway, and much less helpful than a 4 or 5.
The reason for b) is that removing 7s, 8s or 9s reduces the chance of the player busting from 16 (encouraging hitting) but simultaneously increases the dealer's chance of busting (encouraging standing). Removing a 7, in particular, has a big impact on the dealer's bust probability. Partly this is because the dealer is less likely to have a 7 in the hole, and partly it is because a 7 won't bust a dealer total of 14 or less. Similar, though weaker, effects are present with the 8 and 9.
In terms of c), removing a ten makes the player less likely to bust (encouraging hitting). The impact on the dealer's bust probability is miniscule. While a preponderance of tens makes it more likely that the dealer will bust from stiffs, it is also more likely that the dealer will have a ten in the hole.
As d) suggests, 5 is the most important small card (followed by the 4) and 6 is the key big card. The reason the EORs for these cards are so big is that the effects on player hitting and dealer busting reinforce each other. Removing a 5 makes it less likely that the player will improve the hand by hitting (encouraging standing) and more likely that the dealer will bust (encouraging standing). In contrast, removing a 6 makes it less likely that the player will bust (encouraging hitting), and less likely that the dealer will bust (encouraging hitting).
To summarize the story for 16 v T, there are only two cards that can significantly help the player if hitting, and the removal of cards that make it less likely for the player to bust usually also increases the chance of the dealer busting.
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The EORs for 16 v 7 show the following:
a) The EORs for aces, 2s, 3s, 4s and 5s are large and negative.
b) The EORs for 6s, 7s, 8s and 9s are large and positive.
c) The impact of removing a ten is negligible.
In terms of a), removing an A-5 makes it more difficult for the player to improve the hand through hitting and reduces the dealers chance of busting, except in the case of the 5, where the effect is very close to zero but of the opposite sign (this explains why the EORs for the 3 and 4 are larger in absolute value than the EOR for the 5).
Turning to b), removing a 6-9 makes it less likely that the player will bust (encouraging hitting) and less likely that the dealer will bust (enouraging hitting).
Regarding c), a preponderance of tens makes a player bust more likely (discouraging hitting), but actually makes dealer busting less likely (encouraging hitting), though the latter effect is only moderate. The impact on dealer busting is only moderate because contradictory effects are present. On the one hand, lots of tens make it more likely that the dealer will have a pat 17. On the other hand, the dealer will be more likely to bust stiffs (for which the 5s-9s play a role).
To summarize the story for 16 v 7, there are five cards (A-5) that can significantly help the player who hits (encouraging hitting). On the other hand, most cards that bust the player also help to bust the dealer (encouraging standing). The first point partially explains the greater benefit of hitting 16 v 7, relative to 16 v T. Nevertheless, the latter point suggests that standing will sometimes become correct. Why, then, is correct standing (from the player's perspective) so rare? To completely answer this, we need to address another issue.
