September 24, 2004

Curious George: Blackjack Decks The house advantage in blackjack increases with the number of decks. How is this? [MI]

These odds assume the player is not counting cards. It's intuitive that card-counting is easier (and/or more effective) with less decks. So what am I missing? Wouldn't one deck have the same ratio of cards as one two or six decks? Here is a FAQ answer about this question saying that busting is easier with fewer decks, and that blackjacks are less likely. Their example shows a hand making 18 with 9 2's, which is obviously impossible with less than three decks. Is this all there is to it? How are blackjacks less frequent with each new deck adding 4 aces and 16 10-point cards?

posted by hutta 19 years ago
  • I think the answer, and I'm not going to think about this very much because my prob and stats is rusty, is that you don't get to use straight ratios to determine probability. You get to use combinations, in the case of blackjack, since order doesn't matter. The trick is that combinations aren't based on x cards out of y total cards, but rather your chances of choosing those cards. So, the probability of getting any given two cards out of a deck of fifty-two are 52 x 51 = 2652. However, there are only 64 out of 2652 combinations of cards that will give you blackjack, so that's a 1 in 43.44 chance of getting blackjack. However, if you have 8 decks, that's (416 x 415) = 172,640 possible combinations of 2 cards, with a total possible of 8 x 64 or 512 combinations of cards that will give you what you need. 172,640 / 512 is a 1 in 337.19 chance of getting a blackjack. (Incidentally, if you're not following the first step, if you have 52 cards in a deck, the chances of getting any given card is 1 in 52. The second card is a 1 in 51 chance, because one card is missing. 52 * 51 is the chance of getting any two random cards.) If that's not completely correct, it should be mostly correct. I welcome any additions or clarifications.
  • Okay, I got a D+ in prob. And that was because they used multiple choice anwsers & I was lucky. I think Sandspider is basically right, but his numbers are way too high. Bear with me. There are N(N-1) ways of dealing 2 cards out of N cards. However, order matters. Therefore, there are two ways of getting a black jack: - Ace + (JKQ10) - (JKQ10) + Ace In single-deck play, that amounts to.. 4 Aces * 16 tens + 16 tens * 4 Aces ------------------- 128 ways of getting a black jack. and 52 * 51 = 2652 ways of dealing two cards. therefore, p1 = 128 / 2652 = 0.048265460030165915 In 8 decks plays, this becomes.. 8 decks * 4 aces + 8 decks * 16 tens + 8 decks * 16 tens + 8 decks * 4 aces -------------------------------------- 8192 ways of getting a black jack with (8 * 52) 416 cards, so 416 * 415 = 172,640 ways of dealing two cards p2 = 8192 / 172,642 = 0.047451343836886005 So it's 1 in 20.71875 for one deck, and 1 in 21.07421875 for eight, or a 0.08 % difference.
  • Waaaaaay too complicated, and I aced statitistics (oh, the wasted time. Here is the easy explanation (from one of my classmates in college who essentially put herself through one of the most expensive colleges in the land based upon counting cards - profs fronted the cash): the count just gives the player a better idea of how many dangerous cards lie in the deck. The dealer has rules to follow and can not hit himself at will, or more importantly, deny hitting himself. When the deck is stacked with high cards the danger to the dealer of hitting himself is higher thus the odds tip towards the players. With fewer cards there is much more chance of the deck getting really out of balance and favoring the players. With more cards the deck can get out of balance but is not as likely to get too far out. The players can still get an advantage, but it is not as great. If the player is not counting, the odds really do not change based upon how many cards are in the deck - well they do, but the player is unaware of whether the odds are favoring the dealer or the player.
  • To quote a great philosopher: "You got to know when to hold
  • I think you nailed it, Richer. Thanks. (Thanks also to Sandpiper for the point about combinations.) Caddis: The claim is that the house advantage increases with extra decks even when there is no card counting.
  • Time lost to shuffling is another reason casinos use multi deck shoes for blackjack. With 1 deck blackjack, the deck needs shuffling far more frequently than an 8deck shoe. More decks = more hands per hour = more money for casino This + increase in house edge + increased difficulty for card counters = multi deck shoes in every casino!
  • This is a clique for those of you who basically don't like mathematics, or any subject related to mathematics, such as chemistry, physics, accounting, economics, etc..! If math isn't your friend either, please join and let the hatred for math spread far and wide >D! This Misha (from PitaTen) layout is brought to you by Hikari - can't you just relate to the girl there? ^_~