Although i never played BJ w/ dealer having hole cards. I'm curious to know how come the 10 in the hole theory is not a good theory but in some books it tells us to assume it's a 10 in the hole. this is very important even if the game where the dealer has no hole card... we shouldn't even assume what the dealer has as a second card right? unless you count? Shout outs to all comments... .appreciate it :wink:
Dealer has a 10 up. You have a total of 19. If you believe the dealer has a 10 in the hole, then the correct play is to "hit" since your 19 will lose to his 20. If you don't really believe he has that 10, then the correct play is to stand since a 19 has only two good hit cards to improve it, and 11 that will bust it.
The "theory" comes from the probability of the hole card being a 10 being 4/13, which is higher than any other single card probability. But 4/13 is hardly a sure thing.
I think your theoram is an off-shoot of this one. If the dealer is showing a 6 or less,there is a 12/13 chance he will take a third card. There is a 4/13 possibility that his second and/or third cards will be a 10,so in the majority of cases,when a dealer has three cards,one will be a ten. Thats why people say to assume that the dealer has a ten when his face card is small. In reality they are assuming that one of his next TWO cards will be a ten. As has been pointed out,if the correct play was to assume the dealers hole card is always a ten, then insuring an ace would be part of BS.