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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