An Ace or Ten as Your First Card

 
I was playing at Casino du Montreal when something very unexpected happened. I had 16 and the dealer had a face card. The count was mildly positive, when something unexpected happened. The player beside me signaled for a hit on my hand! Immediately, the dealer gave me an ace.

I protested. After all, I hadn't signaled for a hit. The policy at Montreal at the time, and still is as far as I know, is that the card is not burned. It just becomes the next card to be played. I stayed on my hand, so did third base with 19. The dealer turned over a face card for 20. The ace was just sitting there, face down in the dealers hand.

I hopped over to first base and put down a chunky $650 bet, and the dealer gave me the ace, and a jack of spades to go with it. Vingt - y - un. Winner winner chicken dinner.

Knowing you will receive an ace as your first card is a big deal in blackjack. It gives you an advantage of 42.08%. That's huge. The average card counter has about a 1% edge on total action. So how much should you bet? You want to bet as much as possible to maximize your EV, but at the same time you have to consider variance. An ace is not a sure winner. You need to bet some % of your bankroll (presumably less than 42.08% as per the kelly criterion). But how much exactly?

One important question is whether or not you will want to resplit. If you are allowed to resplit aces, you might want to hold some back. That's pretty rare (1 in 13), but is huge when it happens. The chance of splitting to three hands is about 1.1% so isn't that important in terms of bet sizing. Whether you should double or not (or even double for less) is a more complicated question.

Grosjean, in his seminal work on the topic 42.08%, offers the following doubling chart:

With a ten, at least the answer is relatively straight forward. You don't have to ever double or split, so you just have your 13% edge to consider. Probably you want to bet anywhere up to about 10% of your bankroll in that scenario.