20 replies to this topic

[Yoruichi-san]

I remember hearing from college-mates that some grad student did their thesis on Blackjack and proved that there was a 51% chance of winning if the player counted cards.

Let's look at a simpler variant, with only one decision making node.

The game is thus: there are 3 players, and a first card is dealt to each, then a second card is dealt to each, then a third card, etc. The game ends when either one player has 20 or 21, in which case they win, or when 2 players have busted with 22 or above, in which case the remaining player wins.

Your decision: which seat do you sit in? For simplicity, assume the cards are dealt from an infinite stack of decks.

[Rob_Gandy]

Spoiler for Well

I just might take the first seat. I'd have about a 15.5% chance of winning after getting two cards. Just a first thought. Still considering numbers.

[bonanova]

Does the game end the moment the cards in one hand say so?
Are the groups of three cards dealt simultaneously?
Else how does seat position come into play?

[CaptainEd]

Must be Bonanova is right. Cards are dealt one at a time, and completion criterion is tested each time. As soon as the first person hits 20 or 21, or the second person busts, the hand is over.

[CaptainEd]

Is an ace either 1 or 11 at the player's choice?

[CaptainEd]

"infinite stack of decks"...One shuffled deck offers sampling WITHOUT replacement, which is what makes it possible for someone to benefit from card counting. If we stacked another shuffled deck on top, and another, etc. producing an "infinite stack of decks", there would still be sampling without replacement. (Indeed, with only three players, you're sure to exhaust the players before the first deck of cards.)

Is the purpose of this phrase to imply sampling WITH replacement?

[~andy~]

You will have no advantage to counting cards if you have got to play to 20/21 or bust
If you are counting cards you need to be allowed to 'stick' many times

I think what is meant by 'infinite stack of decks' is that many decks get shuffled together (as in the proper casino game, usually six), not one deck after another

Therefore which seat you take does not make any difference whatsoever, card counting cannot come into it

[TheChad08]

You will have no advantage to counting cards if you have got to play to 20/21 or bust
If you are counting cards you need to be allowed to 'stick' many times

I think what is meant by 'infinite stack of decks' is that many decks get shuffled together (as in the proper casino game, usually six), not one deck after another

Therefore which seat you take does not make any difference whatsoever, card counting cannot come into it

The question is about probability of reach 20,21 or 22+
If the odds are higher than you'll reach 20 or 21, you want seat 1.
If the odds are higher that you'll bust, you want seat 3.

1/13 chance of an Ace
1/13 chance of a two
1/13 chance of a 3
etc.
1/13 chance of a 9
4/13 chance of a 10 card (10 or face).

Using those numbers, what are the odds of being dealt a 20, 21 or 22+?

[Rob_Gandy]

I'm gonna stick with seat one. I'd have about a 15.5% chance to win after getting my second card. Then I have around a 55% chance to not bust on my third card. I did it all on paper so I can't really go much further without getting the computer into it. I may try to do this as I have time, but I'd have to do it on excel because that's all I have at work and I wouldn't know how to write a program to do it for me.

[Yoruichi-san]

Does the game end the moment the cards in one hand say so?
Are the groups of three cards dealt simultaneously?
Else how does seat position come into play?

Yes, cards are dealt one at a time.

Is an ace either 1 or 11 at the player's choice?

It will count as whichever will make the player win or not lose if there's a difference, otherwise it doesn't matter.

"infinite stack of decks"...One shuffled deck offers sampling WITHOUT replacement, which is what makes it possible for someone to benefit from card counting. If we stacked another shuffled deck on top, and another, etc. producing an "infinite stack of decks", there would still be sampling without replacement. (Indeed, with only three players, you're sure to exhaust the players before the first deck of cards.)

Is the purpose of this phrase to imply sampling WITH replacement?

Yes, it's to imply, say, the probability of getting an A on your next turn is always the same, regardless of what cards have already been dealt, for simplicity.