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Analysis

This page will give you some insight into how seemingly simple statements can provoke complex discussions and analysis behind the scenes amongst OZmium analysts and commentators. The original query was raised in the section 'Probability' in the subsection 'If only it weren't for that player on the end'. The people discussing this have backgrounds in areas such as pure mathematics, probability and actuarial studies and should know what they're talking about!


Guy.

Went to have a look at Smartgambler. Reading through the section on probability theory I noticed the following commentary on blackjack.

"Except for rare cases where the player has special information about the deck through card counting, the likelihood of the remaining cards in the deck being beneficial to the dealer is completely independent of what the last player does. Sometimes it will be good, other times bad."

Hmm, not sure about that theory. I would suggest that for the current hand, whether the anchor man does or doesn't take a card will not impact on expected return or the probability of the other players winning. It may of course impact on his own chances.

Given any mix of cards remaining (even in the example of a high count), if he hits and gets a low card, the odds of high card next are increased. If he gets a high card then the odds are decreased, but on a weighted probability (between high / low) this is equal to the increase in odds if he gets a high card.

I think the following example will demonstrate this simply. Suppose I have a full deck of cards plus one joker (ie 53 cards) and a room of 53 people, deal the cards out and offer a prize to whoever gets the joker. Do you want to be dealt the first or the last card ? Or somewhere in between?

Of course probability says it makes no difference. The chance of the first person is 1 in 53, the second person is also one in 53 which is 1/52 x 52/53, (ie the odds of them getting the joker conditional upon {ie times} the first person not getting the joker)

Maybe some food for thought? I think this logic applies to the anchor man's decision making.

Equally, I think that it can be taken one step further and applied to the odds starting the next hand, as his choice is equally likely to have a proportionate increase / decrease to the card count at the beginning of the next hand.

The only way that it could reduce the expected outcome in a high count shoe would be if taking an extra card ended the shoe one hand earlier.

Regards, Adrian.


Hi Sean.

Adrian (and Neil backed him up) seem to disagree with something in our Blackjack section, originally written by you.

The trouble is, I've read what Adrian sent me and agree with his analysis, but I don't quite see how it contradicts what you wrote. I'm not going to change anything unless I understand what it is that is supposed to be wrong. Can you read his e-mail and see if you can figure out what his contention is and tell me if you think there's any problem with what we've got on Smartgambler? I'm baffled.

Thanks, Guy.


Howdy Guy.

Adrian's point is that the independence of the dealer's cards also holds true regardless of what information a card counter has. In other words, the statement 'except in rare cases' should be removed.

The problem Smartgambler has is that it must protect itself against pedants who will say 'but when you said this, if this occurs and under this circumstance and if this was known...then you're wrong!' So we put in footnotes or asides to cover ourselves against theoretical possibilities.

In this specific case, for example, Adrian is completely correct in principle. However there are (and I used the word 'rare' in the article), theoretical cases when it is possible to affect the likelihood of the outcome of the dealer's cards by the anchorman's decision whether to play or not.

The principle Adrian rightly brings up is that even by knowing exactly which cards are remaining in the deck, because we don't know what order they're in, we can't change the odds of the dealer getting any particular one of them. So the anchorman deciding to take a card is the same as deciding not to as far as the distribution of the dealer's cards. For example, suppose you know the next two cards are a five and a ten (superb counting in a single deck), and you are deciding whether to take a card or not. Because you don't know which card is next, you are a 50% chance of getting either. If you get the five the dealer has 100% chance of getting a ten and if you get the ten he has a 100% chance of getting a five, the average of which is 50% of getting either, the same as if you decided to sit! In other words, taking a card has no effect.

Only by gaining knowledge of the ordering of a sequence of cards can you make a decision to change the likelihood of the next card. If we know the count is high or low, for example, then hitting near the end of the deck has the effect of increasing the likelihood of the dealer having to reshuffle and hence re-set the count to zero. However, in real life the shoe is always cut so that the dealer never needs to shuffle again for his own cards, so this doesn't count. If there was a game (perhaps one deck in Vegas), where this occurred, then the anchorman could indeed affect the outcome.

The only real method I can think of that would work is by shuffle tracking. If the tracker knows that the distribution is getting ready for a change then he can have some small impact on the distribution of the dealer's cards by his decision. Again, this impact would be quite small and is only a theoretical point.

Unfortunately, being pedantic myself makes me loathe to allow another to pick holes in a statement. For example, if someone asks me what path a ball takes when thrown (ignoring local effects and air resistance), I say an ellipse rather than a parabola, because strictly speaking the ball orbits the centre of the earth and is more accurately predicted by an elliptical equation and only approximated by a parabolic one!

Of course, even then I have to consider whether to renounce the ellipse in favour of the path of a geodesic in Einsteinian space-time, which is considered a more accurate depiction of events than the Newtonian elliptical one. The fact that the elliptical equation would end up making only a micrometer of difference compared to the parabola and that Einstein's would make about a trillionth of even that, doesn't matter!

Of course the fact that real world effects so monstrously outweigh these pedantic classification differences by so many orders of magnitudes really makes the whole thing a moot point. But I digress...

In reality, there is really no practical way even a card counter can impact things, so the caveat should probably be removed from the statement to avoid complicating the issue. The principle of the dealer independence is the important point and certainly of most relevance to the SG readership.

Otherwise, a discussion about these more abstruse points could be moved into a footnote with a detailed description like the above in it. In fact, I'm sure Matt would love to set up a user feedback section at the bottom of each page where users can post comments (much like a forum), about each article and the discussions like we have entertained could be put in for readers to indulge in if they desired! Let's face it, the Hegelian dialectical process of refinement of ideas by staking two poles in the ground and coming to something new and clearer in between is well reflected by these sorts of feedback mechanisms.

Sean.


Okay, here's an hypothetical situation.

A 7 deck game and you are under the gun. The count is very high and there's half a deck left. (Okay, it's not going to happen but it's to illustrate a point). You are sitting on 12 versus 6. The count is very high. The other 6 players all have soft 20.

Now let's assume they all decide to double down. With each card taken, the count is likely to fall. The count is 'mean-reverting', it always 'tries' to get back to the mean, which is zero. By the time the six players have all taken their cards, the count is very likely to be much lower than when they started. This makes a large difference to your expected return on this hand. You might be pretty pissed off, unless you happened to win anyway. Every card drawn affects the count. Every extra card taken by other players is likely to change the count towards the mean, (0). The effect is usually negligible but it does exist theoretically.

Cheers, Matt.


Although the count is much more likely to be lower by the time it gets around and the dealer thus more likely to score a lower card, IF the count isn't lower then the dealer's PR of scoring a higher card is increased commensurately, in fact coming out exactly as if the players didn't bother to take any cards.

The easiest way to view this is to assume that we know exactly what cards are left in the deck, just not what order they're in. Imagine there are nine 10s and one 2 and the count is thus very high. Although we know that it is likely that the count will reduce as the cards are taken, IF the 2 comes out then the dealer gets a 10 with 100% certainty. If you add up all the conditional probabilities at the outset of the decision making, the dealer should end up with a 9/10 chance of getting a ten and a 1/10 chance of getting a two, no matter which of the players plays or stands. It is only after some or all of the players have made their decisions that the gun player might decide that things look different for him, but at any particular stage in the proceedings no player can make a decision to influence the dealer.

In other words, the players are only acting to shuffle the remaining cards in the shoe as far as the dealer is concerned, they can't actually reduce or increase the probability of what the dealer gets. Once one or more cards have been dealt, a player may feel good or bad based upon the contribution of that card to the count, but prior to it being dealt, unless he has knowledge about how the cards are ordered in the remaining deck, he shouldn't care one way or the other about the next player's choice to hit or stand, as the probabilities should come out in the wash to be exactly the same.

Sean.


Matt, unfortunately I still hold a different opinion. Maybe the following explanation may bring our thoughts together.

Although it is likely that the count will be lower by the time you get to draw your cards, there is also the possibility that the count will be higher by the time the deal gets to you. The probability of the card count being worse multiplied by the deterioration in the deck is offset by the probability that the deck improves multiplied by the level of improvement.

To expand upon your example, assume one half deck of 26 card containing 25 picture cards and a single "5". For the players (by taking or not taking cards) to influence the outcome of the hand, they need to alter the probability of the dealer getting the "5" card in which case the dealer would win.

If no-one took a card, clearly the dealer would have a 1/26 chance of winning. If one card was taken prior to the dealer the probability of the dealer getting the "5" would be the probability that the player didn't get the "5" multiplied by the chance that the dealer does from the remaining cards, ie. 25/26 x 1/25. Still 1 in 26.

If 3 cards were taken before the dealer got his card the probability of the dealer winning would be calculated in the same way, i.e. 25/26 x 24/25 x 23/24 x 1/23, which is still 1/26.

Matt, I do however appreciate that in the scenario you have provided that the players may be able to increase their expected returns if they split in this scenario, ie by betting more when the odds are clearly in their favour (which is the whole idea behind card-counting), but I do not believe that they can influence the underlying probability of the players winning or losing the hand.

Lots of food for thought.

Regards, Adrian.

PS. If the four of us do not immediately find concepts such as this as crystal clear, what chance does "Joe the punter" in the casino have?


That'll teach me to argue intuitively about probability without doing a calculation. You'd think I'd know better by now!

Getting back to Sean's reply. I wouldn't be that surprised if there were one or two people on the planet who could track all the cards through 8 decks.

Acording to the physicist Murray Gell-Mann, one of the few times James Randi failed to debunk someone who claimed to have special abilities was some guy who claimed he could name the composer, piece, and sometimes conductor and orchestra of any classical music on LP by looking at the grooves in the record!

Matt.

(Editor's note: The paragraph above is not quite accurate. We were contacted by James Randi who gave us the correct version, as follows. We apologise to Murray Gell-Mann and James Randi for the mistake.)

"The matter Murray referred to was an investigation I made for TIME Magazine. There was no paranormal claim made and TIME just wanted me to discover whether there was any trick. There wasn’t. The guy could identify classical LP recordings just by examining the size of the 'cuts', the dynamics of the grooves and other physical attributes. He was bang-on with 20 recordings I tested him with, including identifying one as “not classical, just garbage.”


I imagine if automatic shuffle machines weren't too good, that someone could come up with a 'clumping' theory that tracks sequences of cards, e.g. 10+4 occurs more frequently than 4+10 and so on, enabling some fancy counting to take place. In this scenario, some ordering information would be available and decisions could be made to change the dealer's cards. With seven people at a table, a ring-in anchorman to 'do the right thing' when they're all sitting on 12 versus a dealer's 6 by drawing or standing when he normally wouldn't might be a feasible scenario!

Sean.


 

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