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Thread: Need help to understand Throp's paper.

  1. #27
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    Quote Originally Posted by peterlee View Post
    For a seven-spot BJ table, dealer's hole card is the 16th card dealt. Chance of getting a picture is 4/13.

    For a no-hole-card game, dealer's hole card would be the 16th or later, due to the players hitting or taking other actions.
    Since it is more often end up with a big card on hitting, the chance of the dealer's hole card being a picture is not exactly 4/13.
    Your first statement is correct if you have no other information about the first few cards dealt to the other players: the probability in that case is indeed 4/13. Unless the dealer's hole card is dealt first from the top of the shoe (which never happens anyway), the probability of the dealer's hole card is never 4/13, but slightly more or slightly less depending on the cards that were dealt first. In other words, the dealer's hole card is conditioned on the cards that were dealt prior to that. You're implicitly thinking about the events as if they were unconditional when in fact they should be recalculated every time a new card is dealt and the information accounted for.

  2. #28
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    Quote Originally Posted by JohnGalt007 View Post
    Your first statement is correct if you have no other information about the first few cards dealt to the other players: the probability in that case is indeed 4/13. Unless the dealer's hole card is dealt first from the top of the shoe (which never happens anyway), the probability of the dealer's hole card is never 4/13, but slightly more or slightly less depending on the cards that were dealt first. In other words, the dealer's hole card is conditioned on the cards that were dealt prior to that. You're implicitly thinking about the events as if they were unconditional when in fact they should be recalculated every time a new card is dealt and the information accounted for.
    Your reply only applies to the first statement, correct?
    [For a seven-spot BJ table, dealer's hole card is the 16th card dealt. Chance of getting a picture is 4/13.]
    I mean, we are estimating the chance before the game starts.
    I'm bringing this up because you mentioned "exchangeability," and I watched some YouTube videos about that. Without any other information, the first 16 cards from the shoe are exchangeable. Is that correct?

    Regarding the second statement,
    [For a no-hole-card game, dealer's hole card would be the 16th or later, due to the players hitting or taking other actions.]
    We are also estimating the chance before the game starts. We don't have any extra information after any cards are dealt. However, for basic strategy (BS) players, some additional cards will be dealt before the dealer receives the hole card. So, [Since it is more likely to end up with a high-value card when hitting, the chance of the dealer's hole card being a picture card is not exactly 4/13.], and I would add "less than 4/13."

  3. #29
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    Quote Originally Posted by peterlee View Post
    Your reply only applies to the first statement, correct?
    [For a seven-spot BJ table, dealer's hole card is the 16th card dealt. Chance of getting a picture is 4/13.]
    I mean, we are estimating the chance before the game starts.
    I'm bringing this up because you mentioned "exchangeability," and I watched some YouTube videos about that. Without any other information, the first 16 cards from the shoe are exchangeable. Is that correct?

    Regarding the second statement,
    [For a no-hole-card game, dealer's hole card would be the 16th or later, due to the players hitting or taking other actions.]
    We are also estimating the chance before the game starts. We don't have any extra information after any cards are dealt. However, for basic strategy (BS) players, some additional cards will be dealt before the dealer receives the hole card. So, [Since it is more likely to end up with a high-value card when hitting, the chance of the dealer's hole card being a picture card is not exactly 4/13.], and I would add "less than 4/13."
    Yes, it sounds like you got everything right. I was referring to just your first statement earlier.

  4. #30
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    Quote Originally Posted by JohnGalt007 View Post
    Yes, it sounds like you got everything right. I was referring to just your first statement earlier.
    Wait a minute. In Thorp's example, even red cards are used more, the chance is still equal in the next round.

    So in this no hole card case, I would say it is still 4/13 on average.

    Does the book "The Doctrine of Chances" offer any methods to solve problem like this?

  5. #31
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    Quote Originally Posted by peterlee View Post
    Wait a minute. In Thorp's example, even red cards are used more, the chance is still equal in the next round.

    So in this no hole card case, I would say it is still 4/13 on average.

    Does the book "The Doctrine of Chances" offer any methods to solve problem like this?
    Looking back on what you were trying to say in your second statement, it's clear that, thanks to Thorp's theorem in the paper, it doesn't matter what the strategies of the players are, the probability of the dealer dealing himself a ten-valued card assuming no other information about the other cards is still 4/13. I wish I could help further, but it seems we are talking past each other with different implicit assumptions. I strongly recommend you pick up the book and read every piece of mathematics in it you possibly can, or just re-read the paper.

  6. #32
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    Quote Originally Posted by JohnGalt007 View Post
    Looking back on what you were trying to say in your second statement, it's clear that, thanks to Thorp's theorem in the paper, it doesn't matter what the strategies of the players are, the probability of the dealer dealing himself a ten-valued card assuming no other information about the other cards is still 4/13. I wish I could help further, but it seems we are talking past each other with different implicit assumptions. I strongly recommend you pick up the book and read every piece of mathematics in it you possibly can, or just re-read the paper.
    Thorp's example:
    [Randomly shuffled deck of two red and two black cards. There are six equally probable sequences. Draw until first red card appears.]
    (1 is red, two rounds played)
    1 1 0 0
    1 0 0 1
    1 0 1 0
    0 1 1 0
    0 1 0 1
    0 0 1 1

    In this example, the red and black cards have an equal chance of being drawn first. However, the rule of drawing until the first red card appears means that the red cards are more likely to be dealt after that.

    In a blackjack no hole card game, the first 15 cards for 7 players and the dealer's up card are equally likely to be any of the 13 ranks. However, the middle part of the game, when players are hitting, more picture cards are used on average. After this, the dealer's hole card is dealt. It is not clear whether this card has a 4/13 chance of being a picture card, or less.

    The paper states that the expected value for different rounds is the same. However, this does not necessarily mean that the probability of a picture card is 4/13 in all positions in the segment of cards for a round played. ...as the red cards in the example.

    It seems to me that there are both reasons for and against the probability of a picture card being 4/13 in all positions in the segment of cards for a round played.

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