While sitting in the car with nothing to do, I was struck with the sudden realization that a player at 3rd base in a game with no dealer hole card has the strange opportunity of choosing whether the next card dealt will be his hit card, or the dealer's second card. In normal games (with a hole card) the thought of "taking the dealer's bust card" and stuff is pretty insignificant (or worthless), since you don't yet know what total the dealer already has due to that unseen hole card. Without a hole card though, you know *exactly* what the dealer already has, and can ask yourself if the next card is *probably* more likely to help you or the dealer.
For example, let's say the player has a 15 (no surrender available) and the dealer has a 10. One of the reasons BS tells us to hit is that we assume the dealer ALREADY has an 8/13 chance of having a 7-A (if the dealer didn't peek), plus the chance to make their hand when they hit. Pretty ugly odds, so you'd risk busting in order to possibly improve your hand. When the dealer has no hole card, however, those same cards that will cause you to bust your hand will also be the cards that lets the dealer make theirs (player busts if a 7-10 comes next; standing means the dealer has 17-20 and wins). Suddenly it has become irrelevent which decision you make nearly half of the time you face this situation -- you lose regardless of your play. Conversely, we may hit on the 15 and get a 3, and be happy with it because the dealer may already have a 7 or 8 under there... and surely getting that 3 couldn't HURT us any, could it? But if the dealer hasn't gotten a hole card yet, could hitting and getting that 3 actually lower your odds of winning the hand? Standing would have given the dealer a 13... This leads to a question that we don't see in American casinos: Is it more profitable for us to have ourselves a better hand facing the dealer 10, or are we better off letting the dealer end up with a stiff hand and hope that they bust?
So, we set up a little table. Let's examine each of the possibilities for the next card in the deck:
A - standing would give the dealer a blackjack... oops
X - player loses either way
9 - ditto
8 - same
7 - and again
6 - standing gives the dealer an 8/13 chance of busting, but I'd feel a lot better hitting and having that 21, and winning 12/13 times.
5 - standing gives the dealer over a 50% chance of busting, while hitting gives the player a better chance of winning (or pushing) the hand
4 - without detailed calculations, I'm not sure which decision is better.
3 - hitting would still give the dealer a 6/13 chance of beating you on their very next card, and then another chance to beat you if they need to hit after that. Standing and giving the dealer a 13 should cause the dealer to bust close to half of those hands. Standing appears slightly better.
2 - Having a 17 rather than a 15 will earn you a push less than 10% of the time, and the dealer still must bust for you to win. I'd rather stand and watch the dealer pull a face card on their 12 4/13 of the time right off the bat.
And just a reminder again: We're ONLY talking about the odds of the 3rd base player; we know that the decisions of the 3rd base player have no impact on the odds of the other players at the table, and I don't want anyone to think that I am implying this.
I don't claim to be any sort of expert on blackjack, but of all the stuff I've read to date, everything assumes that the dealer's hole card is unrelated to what the player might get if he hits. I am curious to know if any studies have been done on this (surely they must be somewhere), and if so, does this quirk of the game warrant any strategy adjustments? My gut feeling is that any changes would only give very small benefits to your expectations. If I knew of an easy way to test this theory I would do it myself, but anything I try would be rather time consuming and prone to error, so some help would be appreciated.
Thanks for your time.
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