Sorry but this is full of errors.
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You should always double down on 11, on 10 when the dealer shows 9 or lower and on 9 when the dealer shows 6 or lower. Always split aces and 8's, and split 3's and 2's when the dealer shows a 4, 5, 6 or 7.
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Firstly you have to establish the rules. If you are playing with a hole card and multiple deck you double down 11 against anything except an ace.
If you are playing single deck blackjack you double 11 against an ace also.
If you are playing european no hole card you do not double 11 against a 10 or an ace. You double 10 against 2 through 9 inclusive in all rules so you got that one right.
Whether you double 9 or not is again rule dependent. For multiplie deck you should double 9 against 3 through 6 inclusive.
You should only double 9 against a dealer 2 in single deck.
You should always split aces except when playing european no hole card when the correct play against a dealer ace is a hit rather than a split with A,A. Also with 8,8 in european no hole card the correct play against a dealer 10 or Ace is a hit and not a split.
Also you don't just split 3s and 2s against a dealer 4,5,6 or 7
Again it is rule dependent and will depend on a few things like are double after splits allowed and are resplits allowed. If they are you would split 2s and 3s against a dealer 2 through 7 inclusive. If double after splits is not allowed then only against 4 through 7 inclusive.
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No matter what the dealer is showing, always assume there's a 10 hidden.
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That is utter garbage. Taken from wizard of odds website....
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Assume ten in the hole: For this strategy I first figured out the optimal basic strategy under this assumption. If the dealer had an ace up I reverted to the proper basic strategy assumption of assuming the dealer did not have a ten. Then I went back and used this strategy under regular playing conditions. This results in a house edge of 10.03%.
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10.03% house edge ? No thanks !
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Since the dealer's up card is so important, assume a 10-value card is underneath before deciding what to do. In a multiple deck game, always spilt aces and 8's, but don't split 4's, 5's or 10's.
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I have covered the corrections to these mistakes above.
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but double down on a soft 13 through 18 when the dealer shows 4, 5 or 6.
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Again that is wrong, of course rules may alter some strategy but if the dealer is showing a 4,5 or 6 you double soft 15 through soft 18 inclusive and you also double soft 19 against a 6. You should only double soft 13 and soft 14 against a 5 or 6. Soft 17 you can double against a 3 and soft 18 you can double against a 2 and a 3 as well.
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There's no need to buy insurance because it's nothing more than a side bet that the dealer with an ace showing has a 10 in the hole. With normal card distribution, the 2-1 payoff is 8 percent against you -- not a good bet.
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You are correct in that you shouldn't take insurance but a minute ago you just told us we should always assume the dealer has a 10 in the hole !
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The only time the odds lean toward you is if you're a keen observer and have a good idea there are fewer than twice as many non-10's than 10's.
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This is one of the basics of card counting but it's waaaaaaaaay more complicated than that.
Please go do a bit of reading before you post such misleading nonsense. People might actually think this is correct and go and lose mney trying it.