Blackjack Math Question

In a 6 deck blackjack game, what would be the average number of face cards and aces dealt per completed round assuming heads up play?

First question: You said face cards. Are you really talking face cards only, jack, queen, king or did you mean 10-value cards, which would include the 10's as well? In heads up action, the average dealer hand is 2.78 cards and the average player hand 2.7 totaling 5.48 cards. So 10-value cards, plus aces, would consist of 5/13 of cards or 38.5%. 5.48 cards x .385=2.1098 or 2.1 per round.

If you really meant face cards and aces (excluding 10's), then the number is 1.68 per round.

I meant to say 10 value cards. :)

So for each added player you would add approximately 1 per round?

That would depend a lot on penetration of the deck. Including 10's and aces there are 20 face cards in the deck. Of course four of these are aces and the remaining 16 10- king. So in a six deck game you would have 96 face cards and 24 aces. If you have a decent penetration that only leaves one deck, then 4 aces and 16 face cards would be left in the deck in theory. Of course with the decks shuffled together your likely to see more face cards than called for in five decks or less. However over enough games I would think that it would average out to the amount of face cards in the deck minus the penetration. So in a six deck game with 5 deck penetration you should see 80 face cards and 20 aces. If I'm off here someone feel free to correct me. I'm still definately learning.


Edit: kewlj beat me to it with a better explanation. He posted while I was typing.

Quote: chrisjs87

I meant to say 10 value cards. :)

So for each added player you would add approximately 1 per round?

Basically yes. That average number of cards for the dealer starts to increase slightly when more players are added. This is because in heads up action the dealer will not finish his hand after the player busts or if the player has blackjack. So with 2 players the average dealer cards goes up to 2.91 and 3 or more players to 2.93. Not really enough to change all that much. If you are estimating, I guess 1 per round, per player, including dealer works as a rough estimate. BTW, these numbers I am using are from Norm Wattenberger's Blackjack in Color chart. I have seen slightly different numbers in other works usually averaging out to 2.8 cards per player (including dealer) per round, but you can't go wrong using Norm's numbers. He's the best.

Quote: Mikey75

That would depend a lot on penetration of the deck. Including 10's and aces there are 20 face cards in the deck. Of course four of these are aces and the remaining 16 10- king. So in a six deck game you would have 96 face cards and 24 aces. If you have a decent penetration that only leaves one deck, then 4 aces and 16 face cards would be left in the deck in theory. Of course with the decks shuffled together your likely to see more face cards than called for in five decks or less. However over enough games I would think that it would average out to the amount of face cards in the deck minus the penetration. So in a six deck game with 5 deck penetration you should see 80 face cards and 20 aces. If I'm off here someone feel free to correct me. I'm still definately learning.


Edit: kewlj beat me to it with a better explanation. He posted while I was typing.

Mikey, I have no idea what you are saying. There will be 20 10-value cards and aces per deck. That is 20 for 1 deck, 40 for 2 deck, 120 for 6 decks. Same ratio. Number of decks doesn't matter.

Also penetration has nothing to do with this equation. Well, there will be more rounds played with better penetration, but that doesn't change anything. Still roughly 2.75 cards per player, including dealer, per round and 38.5% are 10-value or aces.

The only thing that would change these numbers is if you know the makeup of cards already played, but there was no mention of card counting, so I am just using the average number as if we have no knowledge of remaining cards.

Quote: kewlj

Basically yes. That average number of cards for the dealer starts to increase slightly when more players are added. This is because in heads up action the dealer will not finish his hand after the player busts or if the player has blackjack. So with 2 players the average dealer cards goes up to 2.91 and 3 or more players to 2.93. Not really enough to change all that much. If you are estimating, I guess 1 per round, per player, including dealer works as a rough estimate. BTW, these numbers I am using are from Norm Wattenberger's Blackjack in Color chart. I have seen slightly different numbers in other works usually averaging out to 2.8 cards per player (including dealer) per round, but you can't go wrong using Norm's numbers. He's the best.

Now given this information, could a card counting system be developed where a player simply counts the 2 through 6 value cards and subtracts the average of the 10's and aces per round? I have no way of testing this and I'm curious if this would work to any extent.

It'd work, but not as well.

I think I misunderstood the original question. I shouldn't have posted because I am certainly still new to this and I'm learning daily. What I thought the op was asking was how many face cards would likely be seen when playing through a six deck shoe. I wasn't thinking in terms of hands. Less penetration would affect how many face cards and aces are likely to show up during a entire shoe. That's why we seek out better penetration games right? Or I am totally missing this? There is a grand possibility that I'm just confused lol.

Quote: chrisjs87

Now given this information, could a card counting system be developed where a player simply counts the 2 through 6 value cards and subtracts the average of the 10's and aces per round? I have no way of testing this and I'm curious if this would work to any extent.

I am not a 'math geek' as I effectionately refer to many AP with super strong mathematical skills. I have very average, maybe just slightly better than average math abilities. But, I always say that I am not trying to re-invent the wheel. I am happy to use the wheel, in this case, count systems, that others long before me have come up with.

Now that said, what you are proposing seems weak to me. You are more acurately tracking the wrong group of cards, the low cards. The real advantage to card counting is knowing when the deck has more Aces and 10-value cards remaining. If you are going to track one group of cards and offset them with projections, I would count the 10-value and aces and then compare to this average of what should have come out. When you have seen less 10-value and Aces than your 1 per player, per round estimate, you know the remaining deck is 'rich' with high cards, which would be advantageous to the player.

Haha well, I would rate myself somewhere near 'extremely deficient' with math skills, so forgive me if i come across as somewhat dense. I'm having trouble imagining how the inverse(accurate 10's and aces vs low card average) would be more effective than my original idea. Considering that both methods accurately track the effect of removal for 5 of the 13 cards and then factor in an average, why would it matter which way you look at it?

Both the effect of removal and the average used would be identical, only inverse.

Total effect of removal for 2 - 6 = +0.027
Total effect of removal for for 10's/A= - 0.026

A simple google search (which I guess I should have done in the first place) turned up the OPP count. Looks like this idea has already been thought of.

i think the Speed Count is based on something like this? I know you count the rounds played when you're using it...