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Math Help - Probability of Blackjack

  1. #1
    bdj
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    Probability of Blackjack

    What are the chances of getting 21?
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    Site Founder Math Help's Avatar
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    This is a complicated question but I'll simplify the answer.

    If you are talking about getting 21 on the inital deal of two cards then that is a fairly easy calculation. You need to first find the chances of getting a 10 card and then find the chances of getting an 11 card.

    10 card

    out of 52 possible cards there are 4 each of: 10s, Js, Qs, Ks which means you have 16 ways of getting 10 out of 52 cards.

    11 card

    out of 52 possible cards there are only 4 kinds of Ace so 4 out of 52.

    Then you multiply the probability of getting each card and you have your answer: (16/52) * (4/52) = 64/2704 = 2.37%

    The answer can get more complicated when you start to think about how there is only 51 cards left after you take the first card etc.
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    Blackjack-mania

    Also dont forget about the possibility of getting 21 with 3 cards or 4 cards or 5 etc.
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    Also, don't forget about the fact that most casinos use multiple decks which makes the odds even worse for you.
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    blackjack formulas

    There is a website with theoretical and applied probability tutorials with a consistent blackjack section, holding some interesting formulas for the probability of being hit favorable cards in the various ensembles of conditions: http://probability.infarom.ro
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    In his text, "Playing Blackjack as a Business", Lawrence Revere stated that in a single deck game one would get a BLACKJACK once every 20.7 hands
    and that for a FOUR-DECK SHOE that would drop to once every 21 hands.

    Revere never explained the discrepancy... I believe Peter Griffin in his book, "Theory of Blackjack" explained the 0.3 discrepancy...

    ..but I don't recall it if he did.
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  7. #7
    Junior Member F.A.P's Avatar
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    Quote Originally Posted by Math Help View Post
    This is a complicated question but I'll simplify the answer.

    If you are talking about getting 21 on the inital deal of two cards then that is a fairly easy calculation. You need to first find the chances of getting a 10 card and then find the chances of getting an 11 card.

    10 card

    out of 52 possible cards there are 4 each of: 10s, Js, Qs, Ks which means you have 16 ways of getting 10 out of 52 cards.

    11 card

    out of 52 possible cards there are only 4 kinds of Ace so 4 out of 52.

    Then you multiply the probability of getting each card and you have your answer: (16/52) * (4/52) = 64/2704 = 2.37%

    The answer can get more complicated when you start to think about how there is only 51 cards left after you take the first card etc.
    It is true that you have 16 cards representing "10" and 4 representing "11" in a single deck of cards.
    To calculate the probability to get "Black Jack" or 21 on the initial deal, we need to consider both events

    A: first card "10" and second "11"
    B: first card "11" and second "10"

    We have

    P("Black Jack") = P(A) + P(B) =\frac{16}{52}\cdot\frac{4}{51}+\frac{4}{52}\cdot\  frac{16}{51}=\frac{32}{663}\approx 0.04827

    where we take into account the fact that only 51 cards remain in the deck after first card is drawn. This is the probability of getting blackjack when using one deck of cards and the deck is "perfectly" shuffled between subsequent deals.
    It corresponds fairly to 1/20.7 as proposed by "Playing Blackjack as a Business" accordning to kahlmy_ishmael_xxiii's post.

    Quote Originally Posted by Juice View Post
    Also, don't forget about the fact that most casinos use multiple decks which makes the odds even worse for you.
    Suppose we have n decks perfectly shuffled between subsequent deals in a one player Black Jack game. Let the events be A and B as denoted above.
    Now

    P(A)=P(B)=\frac{16n}{52n}\cdot\frac{4n}{(52n-1)}=\frac{16}{13}\cdot\frac{n}{(52n-1)}

    P("Black Jack") = P(A) + P(B) =2P(A)=\frac{32}{13}\cdot\frac{n}{(52n-1)}

    With n = 4 we get a probability of 128/2691 or about 0.04757. This is close to the proposed value of 1/21 by "Playing Blackjack as a Business" accordning to kahlmy_ishmael_xxiii's post.

    The situations get more complicated as many hands are dealt after another without shuffling. Also... to find the probability to get 21 with many cards, knowing what cards are dealt, maybe knowing the number of decks, maybe knowing the decks aren't tampered with etc. ...is quite a complicated task...
    Last edited by F.A.P; December 21st 2006 at 11:43 AM.
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    Quote Originally Posted by kahlmy_ishmael_xxiii View Post
    In his text, "Playing Blackjack as a Business", Lawrence Revere stated that in a single deck game one would get a BLACKJACK once every 20.7 hands
    and that for a FOUR-DECK SHOE that would drop to once every 21 hands.

    Revere never explained the discrepancy... I believe Peter Griffin in his book, "Theory of Blackjack" explained the 0.3 discrepancy...

    ..but I don't recall it if he did.
    with a 52 card deck the probability that a random hand is balckjack is:

    2*16*4/52/51 ~= 1/20.7

    If you have an infinite number of decks in the shoe the 51 becomes a 52
    as we are now effectivly doing the dealing with replacement, and the
    probability becomes:

    2*16*4/52/52 ~= 1/21.125.

    Now the 4 deck case is getting close to the infinite case. Doing the
    calculation exactly it is:

    2*(16*4)*(4*4)/(4*52)/(4*52-1) =1/21.023..

    RonL
    Last edited by CaptainBlack; December 21st 2006 at 11:45 AM.
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    Since this is a Black Jack thread, I might as well ask
    What does it mean to count cards?
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    Quote Originally Posted by Goistein View Post
    What does it mean to count cards?
    It means to keep track of some of cards that leave the deck by memory. And then being able to compute the necessarily probabilities. Casinos hate it because people who train themselves to do this do have a small advantage, enough to win money.
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    Quote Originally Posted by Goistein View Post
    Since this is a Black Jack thread, I might as well ask
    What does it mean to count cards?
    It is a ImPrefectHacker said, it is to use a system to keep track of which
    cards have been played so that you can modify the way you play/bet to
    give yourself better odds than you would otherwise have.

    Casinos use multiple decks in the shoe to make this more difficult. Also I
    believe they throw you out if they think you are counting cards.

    A player may well regard card counting as legitimate, but Casinos are a
    business, they cannot make a profit, and in fact can go out of business
    catastrophically if the punter has an edge.

    RonL
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    I saw this thing on the Discovery Channel (I think) where a team of card counters played on a single blackjack table. They said one of the difficult things is the concentration. Because after you start winning a lot people start to watch and all excitment is directed toward you. But they had a lot of practice before they came to Las Vegas. Eventually they got thrown out of the Casinos. I think they made something like 500,000 dollars (I would say in a single night but I do not remember).
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    Dont forget, the odds of you getting your cards, you have to factor in everyone else is given 2 cards, so your odds change as each card is passed out

    Vegas plays with four decks.

    But, the rule i thoroughly enjoy, dealer must hit all the way up to 17. So you've got a good chance of figuring out weather or not the dealer will bust, based on everyone elses cards.

    That is playing 1 on 1 against the dealer, not competition black jack.
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    The most common way to count cards is assigning cards:

    2-6 a +1 value
    7-9 a value of 0
    10,face cards, ace a -1 value

    then when the cards are dealt you either add or subtract according to the cards that are dealt. In theory, the more positive the number, the better chance you have of getting dealt a better hand. Of course the theory has other things like true count, running count, etc... this is just an oversimplified version.
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    Different Amount of Decks Per Casinos.

    Quote Originally Posted by MadSKillzMan View Post
    Dont forget, the odds of you getting your cards, you have to factor in everyone else is given 2 cards, so your odds change as each card is passed out

    Vegas plays with four decks.

    But, the rule i thoroughly enjoy, dealer must hit all the way up to 17. So you've got a good chance of figuring out weather or not the dealer will bust, based on everyone elses cards.

    That is playing 1 on 1 against the dealer, not competition black jack.
    Actually, most places in and out of Vegas use six deck shoes. (A few use four, eight, or even ten.) Most "High-Limit" tables use two decks, and some casinos offer a "Single Deck" blackjack game, but you have a VERY limited amount of hands/places played. (Usually five.) Meaning, that most single deck games only seat 4 players, no "Mid-Shoe" entry, and the dealer's hand counts as a hand/place. Basically, if the table is full, you get one time dealt around then reshuffled.
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