Win a SMALL fortune with counting cards-the math of blackjack & Co.

Win a SMALL fortune with counting cards-the math of blackjack & Co.


Welcome 🙂 Okay this is a bit of a special
Mathologer today. A number of you have requested that I do something on
blackjack and card counting so here we go–how to gamble yourself to fame and
fortune. I am being assisted today by fellow mathematician, longtime colleague
and part-time gambler Marty Ross who is really good at this stuff and who
has offered to share some of the mathematical secrets to coming out on
top in gambling games like blackjack. Okay so let’s begin with a couple of
puzzles. For the first puzzle suppose you’re looking to bet on roulette. The
roulette wheel is numbered from 0 to 37 with 18 red numbers, 18 black numbers and the green 0. So the chances of red coming up is just under 50/50. Now let’s suppose
you’ve been watching the roulette wheel and of the last 100 spins red has come
up 60 times. What should you bet will come up next: red, black, doesn’t matter? Sounds
too easy? Well this probably comes as a surprise but most people get this one
wrong. We’ll give the answer in a little while. Our second puzzle actually arises
in practice–a standard way that casinos and gambling sites sucker people into
betting. For this puzzle you’re given a $10 free bet coupon. You can use the
coupon to place a bet on any standard casino game: roulette, blackjack, craps, and so on. If your bet wins then you receive the normal winnings. For example, let’s
say you bet red on roulette. If red comes up you win $10, of course. Win or lose, the
casino takes the coupon. Now here’s the question: what is the value of this
coupon? In other words, what should or would you be willing to pay for such a
coupon? We leave that one for you to fight over in the comments. But we’ll
give you a hint: whatever you think the obvious answer is you’re definitely
wrong 🙂 Now on with making our fortune. Famously the mathematician Blaise Pascal
sorted out the basics of probability in order to answer some tricky gambling
questions. When not dropping rocks Galileo also
dabbled in these ideas. So if we roll a standard die, then there’s a one in six
chance that five will come up, on a roulette wheel there is a 1 in 37 chance
that 13 comes up, the usual stuff. And then comes in the money. What really matters
to a gambler is not only the odds of winning but of course also how much they
get paid if they win. right? And that is the idea of expectation, the expected fraction of the gamblers bet he expects to win or lose. As an example, suppose we bet a
dollar on red on roulette. We have an 18 in 37 chance of red in which case we
win $1. There’s also a 19 and 37 chance of losing $1. And so, if we keep betting
$1 on red, on average we expect a loss of 18/37 – 19/37 which is – 1/37th of $1, or
-0.03 dollars. What this tells us is that in the long run we expect to have
lost about 3% of whatever we’ve bet. 37 spins and we expect to have lost about
one dollar. 370 spins and we’ve lost about $10 and so on. Of course, dumb luck
can mean that the actual amount we might win or lose may vary dramatically. Again,
in maths we express all this by saying that the expectation of betting on red is –
1/37th or minus 3%. As another example, what if you bet that the number 13 comes
up? If 13 comes up we win $35 and there’s a 1in 37 chance of
that. There’s also a 36 and 37 chance of losing your dollar and so our expectation
comes to 35/37 – 36/37 or -1/37 which as in the first roulette game that
we considered is equal to minus 1/37. In fact, no matter what you bet on
roulette, the expectation will always be – 1/37 give or take some casino
variation. Expectation can vary dramatically on gambling games,
from close to 0% on some casino games down to -40% or so on some
lotteries. But, unsurprisingly, the expectation is pretty much guaranteed to
be less than zero and minus means losing. So far so really really bad 🙂 Hmm
what can we do about it? Well a popular trick is to vary the size of your bet
depending on whether you win or lose. The most famous of such schemes is the so
called martingale. This betting scheme works like this: as before let’s bet on
red in roulette and let’s start by betting $1. If red comes up you win $1
and you repeat your $1 bet. If red does not come up you lose your dollar. To
make up for your loss you play again but this time with a doubled wager of $2.
If red comes up you win $2 which together with the $1 loss in the
previous game amounts an overall win of 2 minus 1 is equals $1. So you’ve won, so you go back to betting just $1. On the other hand, if red does not come up you lose
your $2 which then adds up to a total loss of 2 plus 1 is 3 dollars. You’ve only
lost so far so you play again, but this time with a doubled wager of $4. If red
comes up you win $4 which together with the $3 loss so far means that overall
you’ve won $1. You’ve won and so you revert to betting just $1. On the other
hand, if red does not come up you lose your $4 which then adds up to a total
loss of 4 plus 3 equals 7 dollars. So far you’ve only lost so you play again but this
time with a doubled wager of $8, etc. So basically you keep doubling your bet
until your bad luck runs out at which time you start from the beginning by
betting $1 next then keep doubling your bet again until you win, and so on. As
long as you stop playing after some win, this betting strategy
seems to guarantee you always coming out on top overall. There are many such
betting schemes the d’Alembert the reverse Labouchere. Apparently these
schemes work much better if they have fancy French names, believe it or not. But do bet variation schemes work? Probability questions like this one can
be tricky, depending in a subtle way on our assumptions. The martingale, for
example, obviously works if you happen to have infinitely dollars in your pocket.
But then why bother gambling? And, of course, whatever you do you can always
get lucky but with a finite amount of money in your pocket, what can we expect
to happen? Well, suppose we make a sequence of bets with the same
expectation for each bet, as in the setup we just looked at. Then the total amount
we expect to win or lose is easy to calculate. It’s just E times that
positive number there and if E is negative then uhoh no luck. That brings us to the fundamental and very depressing theorem of gambling. The theorem says
that if the expectation is negative for every individual bet then no bet
variation can make the expectation positive overall. Damn ! 🙂 Okay, so we’re not going to get rich unless we somehow find a game with positive expectation. For the
moment, let’s just assume that such a game exists. How well then can we do?
Suppose we’re betting on a casino game for which the chances of winning are 2/3
and therefore a chances of losing are 1/3. Let’s also assume that just like in
betting on red in roulette you win or lose whatever amount you bet. Then the
expectation for this game is actually positive. To be precise it’s a whopping
33%. Now such a huge positive expectation in the casino game is clearly a fantasy. But
bear with us. Ok, suppose we start with $100. What are the chances of doubling
our money to $200? Well, obviously, if we just plunk it all down in one big bet of
$100 then the chances of doubling are, well, 2/3, of course. This may come as a
surprise but we can actually improve our chances if we bet $50 at a time and we
play until we are either bankrupt or we have doubled our money. Let’s do the maths. If we place bets of fifty dollars, after one bet, win or lose,
we either have 150 or 50 dollars. And after two bets we have $0, $100
or $200. Now, reading off the tree, we see that at this point the probability of
having doubled our money in the first two plays is 2/3 times 2/3 which is
equal to 4/9. And, similarly, the probability to be back to where we
started from with $100 is, well, 2/3 times 1/3 plus 1/3 times 2/3 which
happens to also be 4/9. But if we’re back at $100 we can keep on playing until
eventually we have doubled our money or are bankrupt. It can actually take it while
before this is sorted out, right? Now if D are the chances of eventually
doubling our money in this way, then D is equal to what? Well, 4/9 the probability
of having doubled our money after two bets plus the second 4/9 the probability
of being back where we started from times the probability to be able to
double from this point on. And what is that? Well we’re back to $100.
So the probability is D again. It’s actually quite a nifty calculation when you think about it. Anyway, now we just have to solve for D and this gives that
D is equal to 4/5 which is 80%. And this is definitely a lot better than 66% that
going for just one bet of $100 guaranteed. Repeating the trick, we can
consider betting 25 dollars at a time. This results in an about 94% chance of
doubling our money. In fact, by making the bet size smaller and smaller we can push
the probability of us eventually doubling our money to as close to
certainty as we wish and once we’ve doubled our money, why not keep on
playing to quadruple, octuple, etc. our money. And since we can push the
probability of doubling our money as close to certainty as we like, the
same is then also true for of those more ambitious goals. Even
better the same turns out to be true no matter what probabilities we’re dealing
with. As long as the expectation of the game we play is positive, as in the game
that was played. The very surprising conclusion to all this is our second
very encouraging theorem of gambling. So here we go. If the expectation is
positive, then we can win as much as like, with as little risk as we like, by
betting small enough for long enough. And so, finally, a bit of very good news, right?
Alright, so all that’s holding us back from fame and fortune is finding a game
of positive expectation. For that, of course, we again turn to the game of
roulette. .. Just kidding 🙂 and we’ll get back
to blackjack in a minute. But there are many approaches to gambling and one
factor to keep in mind is that games like roulette are mechanical which means
that the true odds aren’t exactly what the simple mathematics predicts. Is this
sufficient to get an edge on the game? Well I won’t go into that today but in
the references you can find some fascinating stories of people who have
tried to and occasionally succeeded in beating a casino in this way and such
attempts continue to this day. And with that in mind, we’ll now answer
our roulette puzzle from the start. So if 60 of the last 100 spins have turned up
red, then you should most definitely bet on red. Of course, feel free to
disagree vehemently in the comments. Ok so finally on to making our fortune at
blackjack, a possibility made famous in the Kevin Spacey movie 21. Well Kevin’s
out of favour, now so should watch The last casino instead, it’s a much better
movie anyway. For this video we don’t really have to
worry too much about the rules of blackjack, so here’s just a rough sketch.
Now blackjack is played with a standard deck of 52 cards or nowadays a number of
such decks. The goal is to get as close to 21 without going over. All face cards count as ten, the aces count as 1 or 11 the player can
actually choose whichever works better for them. In blackjack you’re playing
against the dealer. You’re initially dealt two cards and the dealer just one, all
face-up for everybody to see. You go first. You can ask for more cards one at a time until you either bust which means you go
over 21 in which case you lose immediately or you stop before this
happens. Then it’s the dealer’s turn who will deal herself cards like a robot
until she hits 17 or above and then stops. The person closest 21 without
having gone bust wins. The casino’s edge comes from you the player having to go
first knowing only the dealer’s first card. So
if you bust by going over 21 then you lose immediately even if the dealer
later busts as well. There are however some compensating factors that favor the
player including the ability to make decisions such as when to stop receiving
cards and whether to “split” or to “double”. We won’t go on to this. Actually the
ability to make decisions only favors the player if they know what they’re
doing which is actually hardly ever the case 🙂 The fundamentals of optimizing
blackjack play involve knowing what decisions to make given any total of
your cards and whatever the dealer’s card and this is known as “basic strategy”
and was actually first figured out in the 1950s by some army guys playing with
their new electronic calculators. The basic strategy can be summarized in a
table which all expert players know by heart. Here’s a simplified version. Let’s
use it. At the moment our cards add up to, well, 10 for the queen plus 5, that’s 15, so
look up 15 on the left side. The dealer has 8 and so the basic strategy
tells us that we should “hit” which means ask for another card. Let’s do that. Now
we’ve got 19 and this means that the basic strategy tells us to stand or stop
which of course makes total sense at this point in time. Figuring out the basic strategy just involves a lot of easy probability tree
diagrams and stuff like that. Casino rules can differ which then changes the basic
strategy slightly as well as the resulting expectation but in a not too
nasty casino the expectation, given optimal play this way, might be
about -0.5%. Close but no banana. Of course plenty of people do worse than that. Casinos play their cards close to their chests but it seems that on average the
casinos make well over 5% on blackjack, a clearly better rate of return for the
casino than on roulette. Anyway, if we want to make our fortune we have to
somehow get around that -0.5% and that’s where card counting comes in. Card counting arose in the early sixties, courtesy of mathematician
Edward Thorp and the fundamental idea is very easy. Basic strategy assumes that
any card has an equal likelihood of appearing next. Well it’s a fairly
natural assumption to make if there’s NO other information to be had but of
course there IS other information to be had as cards get dealt the probabilities
change. In general, high cards are better for the player and low cards are worse.
Then, as the cards are dealt out, the expectation changes and the expectation will be positive if sufficiently many low cards are dealt. That sounds like a lot
of information to keep track of but counting simplifies it all down to
keeping track of just one number called the running count. Every time the cards
are shuffled the running count resets to 0. After the shuffle whenever you see
a low card you add one to the running count. Whenever you see a high card you subtract one. Otherwise you don’t do
anything. The running count indicates how many extra high cards there are among
the cards left to be dealt. Keeping track of the the running count may seem tricky to
do in a casino with all the cards zipping around on the table but it’s actually pretty easy watching a blackjack table for
about an hour most people can keep track of the running count pretty accurately.
There are also plenty of apps around like that one there if you want to
practice in the safety of your home or you can just get a plain old deck of
cards. Now were any of you fast enough to keep track of the running count just
now, over there. I showed this one to Marty cold and he just had it
straight away. Anyway what we really want to know is not the number of extra high
cards left to be dealt but the fraction of extra high cards remaining. For
example five extra high cards matter much less if they’re within three decks
left to be played than if there’s only one deck left to be played. To account
for this we simply take the running count and divide by the number of decks
left to be dealt. This number is called the true count and here’s the surprisingly
simple formula that relates the true count to the expectation at the given
point of the game and this formula contains some really good news. A true
count of two or greater means that our expectation is positive, right two minus
one is positive. A true count of plus ten which can easily happen just before the
shuffle means the expectation is 4.5% which is pretty amazing. So what does the
card counter do? Well, ideally, she bets little or nothing when the true count is
negative, makes small bets if the true count is slightly positive and then
larger bets when the true count is higher. The bad news is that betting in
such a manner involves a lot of boring waiting around followed by frantic and really really suspicious betting perhaps hundreds of
dollars on a few brief hands. How well does it work? Well these days a typical
betting scheme going up to say a maximum bet of 200 dollars might result in an
average of about 15 dollars an hour. Wow, hmmm not what I would call a great
hourly pay. And it gets worse, the result in any given hour can differ massively.
You can expect a standard deviation, a typical plus or minus to be
about $500. Of course the way card counters bet makes them very easy to
spot and Marty has had his run-ins with casinos. So unless you’re part of a well
drilled team of counters and players or you’re really good at disguises there’s
a fair chance you get to meet some burly casino employees within a few short
hours. Well we did say blackjack is a way to win a SMALL 🙂 fortune.
Good luck happy gambling and that’s all for today … Except we’ve all heard that
back in the 70s there were lots of people making millions of dollars
playing blackjack in the casinos. So what has changed? Why can’t we make millions
of dollars these days. (Marty) well the casinos have gotten a lot more careful and a lot
smarter: they use more decks which means the running count matters less, the
true count is slower to get going, they use automatic shuffling machines, they
really are on the lookout for suspicious betting. So unless you’re incredibly good
at disguising yourself, incredibly good at team playing, it’s pretty much dead. (Burkard) It’s dead, that’s sad but what about other games? There’s online gambling now so are there other ways to make money with gambling these days. Absolutely yeah the casino is always
looking to sucker more people into betting and suckering old people into betting more more, so there’s always promotions, there’s new games, new rules,
some are knowingly have expectation which is positive and they just watch
out, others the casino makes mistakes or online betting sites make mistakes. So
you do a little expectation calculation and often not always but often you can
find a little edge and enough of these little edges and you can make a nice
little profit on the side and definitely there’s some people who just
computerized everything, calculate to the nth degree and there’s some secret
people I’m sure who are doing very very well. All right. Well that’s a perfect
lead-in to our next video, at some point. Anyway thanks Marty for coming today.
Thank you and we’ll have you again soon.

100 thoughts on “Win a SMALL fortune with counting cards-the math of blackjack & Co.

  1. Back from an extended holiday. Still working on the ultimate Zeta function video, but to get things rolling again here is a video on the math of casino games. The video is based on talks by my friend and colleague Marty Ross who who knows a lot about beating the casinos at their own game. I do a bit of an interview with Marty at the end in this video but just in case you are interested in seeing him in some real Marty action check out the links in the description 🙂

  2. TY for discussing the probabilities, I'm going to make the safest bet I knovv next time…to just not gamble,as I'll have a 0% chance of losing hard earned money & a 100% chance of having a vveeks vvorth of groceries that payday.

  3. The count is plus 7. It helps if you practice and were taught how to count by working surveillance for casino which I have

  4. ONE problem which illustrates the difference between theory and practice. Just as soon as you begin winning more than their statisticians consider "normal", you will be promptly ejected from the casino. Furthermore, in Los Vegas, the casinos share a database of unusual winners and, once banned from ONE casino, you will be banned from the rest. So, this may work for a couple of hundred dollars, but after that you will no longer be permitted to play at the casino EVER.

  5. Most US casinos use double zero roulette wheels. There is a green '00' opposite the green zero. This doubles the house of advantage.

  6. Casinos these days have automatic card shuffling after each hand, so there goes card counting. There are way more sophisticated counting schemes than the one you present! I know one that has a low, positive E – but you won't be detected. It's also very hard work. 🙁

  7. Great video.
    £10 Coupon is has an EV of about £9.46, assuming someone bets it on a number on European roulette. ($360 – 10)/37

    I've staked £410,000 at about £20 per hand on blackjack (continuous shuffle, no card counting) in the past year, with casino bonuses giving me a +0.5% edge. This video has given me the motivation to play live games + try and get that edge up XD

  8. I do not understand the attraction of gambling at all. Can somebody please explain it? Is it a mental illness? Is it due to a genetic defect? Are gamblers just too stupid to understand the math? Do they believe in premonitions or other paranormal claptrap?

  9. Counting cards is a little over rated unless it's a one deck game. And it's hard for casinos to prove your counting cards.

  10. The best thing to bet on is Football you place your bet and watch the game for 3 hours. And it's always 50-50 on whether your pick is right or wrong. And they only take the vig if you win

  11. I watched it in 5 or 6 jumps,what dahell was he talking about? and why this fascination about some black guy named Jack,was it?

  12. I've tried Martingale betting in simulations and it's worked everytime. In actual practice, casinos have systems in place. (a) you need to stick to your system at all cost (b) casinos have table limits; lower and upper. A $5 minimum table has an upper limit of $299. If you consider your worst case scenario; $5 lost, $10 lost, $20 lost; $40 lost; $80 lost; $160 lost; you now can't do a $320 bet and you just lost $335 dollars.

  13. The dealer has an edge because the player has 2 up cards and the dealer has 1 up and 1 down and if the player uses basic strategy that also adds to the dealers edge.. The player can get a mathematical advantage by not using basic strategy , together with not busting..The player has to also apply certain tactics , also the player needs to look for a table that has the right odds and also a table when the dealer is in bust mode ..The table must have a shoe instead of an endless shuffling machine… When the player doesn't bust , the player has 2 ways to win and only 1 way to lose ( That is the theory at least ) Where as the dealer has 2 ways to lose and only 1 way to win.. However I have yet to test the theory?? I need to get six decks and deal a few thousand hands and apply what I call the 11/12 strategy and see if the flow of the cards favors the player or the dealer??..I dare say some shoes will favor one or the other??.. The test might take a while??..

  14. Card counting at blackjack was a fun and very profitable past time in the 1970s. Since then the casinos have totally wised up.
    If you want to waste hundreds or thousands of your hours in a fruitless and disappointing pursuit, welcome to card counting in the 21st century.

  15. Ok in theory we have 2 methods to win the casino the blackjack(21) or some video poker games who is "profit" to players(e.g wild 2) but really need to remenber tons of combination…

  16. You think $15 isn't a good wage, yet a majority of parole make less than that and US Federal minimum wage is less than 1/2 that, and equal to what progressives think it should be. I'd say counting cards is a great hourly wage.

  17. if casino have a problem with card counting then why dont they just install auto shufflers like in my country??

  18. I think that I read somewhere that it is actually illegal to count cards in the state of Nevada. ls that really true? It would basically mean that you are only allowed to lose money in a casino. Which was the whole point obviously 😀

  19. About the "unfairness" of not "allowing" card counting: With every game of luck, the casino offers a bet. They bet against the player, and usually win in the long run. If they realize, that against a specific player their odds are worse than usual, they will stop offering bets to him.
    The player sure has every right to watch the game to improve their odds, but the casino never abandoned their right to do the same. No one is obligated to offer or accept a bet, when the odds are against him. Not the player, but also not the casino.

  20. I left a reply in the thread below explaining why I think the coupons true value is (upto) $9.45…. I'd love to hear your thoughts.

  21. 1:50 10-10/x approaches 10, so it should be slightly less than 10 (due to a combination of 1) casino games don't go to infinity and 2) the ratio of payout is usually slightly lower than 1/chance

  22. win or lose the casino take the coupon….so it's worth 5$, cuz if i win, i get 10$ and lose the coupon…meaning my bet was worth 5$ + I won 5 = the 10$ I receive for winning……because normally, if I bet 10$ of my own money and I win, I get 10$ + my money back, meaning I would have 20$ in my pocket

  23. This guy is fool of crap. Card counting works, but if you demonstrate that you are consistently making the correct play, the casino simply asks you not to play. You play the game. They have expert counters that review the tapes of your play after you have aroused the casino's attention by winning of a few hours or days. So good luck.

  24. I would love to gamble at a casino where the roulette singleton gives 35 on win… in Sweden it is 30 on win, so the expectation is much less…

  25. Hey I was wondering if you could run the math on this strategy for craps I have no idea how to do it lol I suck at math please help https://youtu.be/STssWCwKHNs

  26. A huge change is blackjack 3 to 2 vs the 6 to 5 on a blackjack and multi decks. It used to shore up losses. I also saw a gambling tip on craps where you just continually bat in this pattern and you don't make a lot of money but you can slowly win money over time. I live in Las Vegas I don't really gamble a lot but I have seen some of the stuff.

  27. Wrong! Even if the red comes up 60% of the time, the probability does not change from 50/50 – red/black. Imagine flipping a coin 50 times. 35 times it comes up heads.When you flip it again, the probability is still 50/50 heads/tails.

  28. Here is my question to who ever knows the ruling. The dealer draw a card. he made 18. But by mistake he draw another card and went 24. What is the ruling here? Do they burn the card like never happen, does the dealer bust, or they cancel that hand? Please help if you know.

  29. You should not expect red next! That's called gamblers fallacy, very interesting for anybody who wishes to google it. 100 spins is NOT enough of a sample to think a table is bias in any way. It just isn't. There are plenty of papers/videos out there that show how often you should expect to see a run of colours in a (almost) 50/50 game. to get 60% from 100 spins is not uncommon at all.

  30. I discovered a system to beat the casinos and would like to share it here, in this comments section, for the first time.
    It all started when I told my mum that I was going to the casino with a few friends to play roulette, she just about hit the roof and told me that you 'can never beat the casino'
    I asked her a simple question: "if you were to place on black or red, and bet 10 pound each time, are you absolutely sure you would lose your money?
    Yes!!! of course I would, she replied.
    I then discovered the perfect system for beating the casino:
    Tell your mum to come to the casino with you and put 10 pound on red and black, and you put 100 pound on the exact opposite!
    You can't lose 🙂 Happy to help.

  31. I would put forward that the value of the $10 coupon would be roughly $9.46 and here's my thinking. 37 coupons are placed on each number (0-36). One number will win for a payout of $350. You now have exchanged 37 coupons for $350. $350 divided by the original 37 coupons gives a total of $9.46. If you can find 36 other people who think the value is less than that and will sell their coupons to you for $9.45 or less you are a guaranteed winner! Wow, finally a system that will work in a casino! Feel free anyone to show me that this isn't a guaranteed win peeps! (And no comments on how the time and effort to find all these people willing to sell their coupons, negotiating the price, doing the exchange, etc are not worth the return. And nothing about the casino only allowing one coupon per person, because that would burst my bubble and just be mean.)

  32. Use card "counting" in blackjack n win. What ALOT of CRAP!!!/bullsh*t!
    The House/casino has a DOUBLE ADVANTAGE over the gambler:-
    Ist Advantage: The gambler draws cards n takes on the risk of NOT busting. If the gambler busts, the casino takes the bet, WITHOUT having to draw any card!
    2nd Advantage: If the gambler draws card n is lucky enough not to go bust, the House/casino ENJOYS the 2nd Advantage of getting the chance/opportunity to draw cards to beat the gambler's hand.

  33. Finally i found a channel were studiying math make sense, imagine that i filed math exams like n times , well i study on part time, buts still to fail math n time is really bad luck , but i still trying to pass the exam 😉

  34. Finally, a video that explains how counting works and shows just how difficult it is to make money counting cards., unlike all those stupid videos that promote a 'winning system.'

  35. I haven't gotten to the answer yet but I'm going to say that if there is some reason (extortion, say) that you can't avoid making a bet on roulette and the wheel has come up red 60 times out of the past 100, then either the wheel truly is random and this is the time that 100 random twirls generated 60 reds (and the odds of that happening are not zero, and they are the same as any other sequence) OR the wheel has a tilt and it's innately more likely to fall red. So bet red. If the wheel is truly random betting red is no worse than betting black.

  36. I've never been able to see why Martingale WOULDN'T work, even if the odds were massively unfair such as an even-money payoff on rolls of a die where the only thing that wins is a "1". With a fair die you would expect to lose 5 out of 6 times. But if you were playing Martingale in THAT game you would STILL come out ahead $1 every time you rolled a win. You would expect to be MAKING those doubled bets a great deal many more times than you'd make in a run of fair coin-flips, but it would still be the case that every time you won your profit would be $1. If you brought a little over 1.5 million bucks to the table the only way you could go broke would be to fail 21 times in a row. That could happen with a die-roll where only one face wins, but 21 losing roulette-throws in a row? Of course the other thing is if the table has a limit. If you're betting at a table where the minimum is $1 and the max is $10, you're going to run up against that.

  37. There's a shoe that kicks out cards randomly from a stack of eight decks. You never shuffle. You just throw the used cards from the last game into the stack. Even if you had 100% of the information as to the order of cards in the stack, that would give you nothing as to odds on the next cards to come out, because they're chosen at random from the ordered stack by a computer. So the only information you have for calculating the odds is the cards face up on that able IN THAT HAND. You can't get any edge (which would be massively hard with 8 decks anyway) from the fact that certain cards are known to have been expended in previous hands. This pretty much kills any hope for a card-counter. At the very least you should go in and fill up all seats at one dealer with accomplices whose mission is to make the smallest possible bets and burn cards until they go bust and lose those small bets, because the higher the number of cards revealed, the more precisely you can calculate what is most likely to come out of the stack next.

  38. The idea that you should bet black on the posed question is the analog of the idea that if a fair coin has come up Heads 9 times in a row then you should bet Tails because "it's due" or "10 Heads in a row, the odds are only 1 in 1024, so Tails is more likely". The fallacy in that argument is that if you flip a fair coin trillions of times, and write down all strings of 10 flips in those results, only 1 out of 1024 of those strings will be 10 Heads. But that would be the problem facing you at a standing start: You'd be a fool to be on the NEXT ten throws in a row to be all Heads, unless the payoff was set to balance the low odds. But that's not the problem you've been given if the fair coin has already given you 9 Heads in a row. You need to look at your trillion coin-flips, divided up into all possible 10-throw sequences, and then throw away all of the ones that did NOT begin with 9 Heads. That mans throwing away 511 out of every 512 of those strings, and culling your sample-size to only 1/512th of all of them. Out of THOSE strings that began with 9 Heads, you will find that 50% of the time the next throw is Heads and 50% of the time the next throw is Tails. 10 Heads in a row from a standing start has odds of 1023/1024 against it, but ten heads in a row when YOU ALREADY HAVE NINE HEADS has a likelihood of 1/2: there's only two possible outcomes: HHHHHHHHHH, or HHHHHHHHHT, each of which is equally likely. The other 1022 possible outcomes shouldn't figure into your calculations because they've already been ruled out as impossible. For instance one outcome you could get in 10 coin-throws is HTHTHTHTHT. But you CAN'T get that outcome anymore if your first 9 throws are already HHHHHHHHH. The HTHTHTHTHT is already impossible at that point, and shouldn't be left as a possible permutation in your odds-calculations. If the coin really is fair then Heads and Tails are equally like for the next throw, but if 9 Heads have come up in a row then you should suspect that the coin is NOT fair but has a defect and is more likely to come up Heads and so THAT is why, rather than unusual luck, you've gotten 9 Heads so far. So you should bet on Heads.

  39. One card-counting system was useful only when a lot of information was known, i.e. many cards had been dealt and the tracking of the remaining cards made it knowable that the house was likely to lose. In this card-counting system the player would make $1 bets for a very long time and then make a huge bet only when the lie of the cards and known information was such that a huge bet would be very likely to pay off. I don't remember the relative size of the huge bet that would have to be made, and won, to cover all of the $1 losses and make it pay. But it was something like $40. The way casinos nuked that strategy was tables where the minimum and maximum allowed bets differed by a factor of 20. If you were allowed to lose all those $1 bets in a row, you wouldn't be allowed to bet $40 when information favored you. Conversely if you were allowed to bet $40 when information favored you, you had to be a minimum of $2, not $1, on all the hands when you were just killing time.

  40. Casinos could easily end all advantages in any card-counting system by having a sufficiently large number of decks dealt by random from a shoe. So why DON'T casinos eliminate all card-counting? It's because they don't play fair with the public. If they configure the games so that card-counting MIGHT work, they make a lot of money from people who THINK they can count cards but really can't do it well enough to win. These people would not play the game at all if they could perceive that the game's structure negated any card-counting strategy. But if you ARE good at card-counting, the casino beats you by barring you rather than by reconfiguring the game in such a manner that would also drive away the amateurs. This is morally wrong. The casino is trying to have it both ways: luring in the amateur losers by telling them "This isn't gambling. It's a game of strategy and memorization that favors you if you have the skills" but then when it finds out you DO have skill saying "You cheated". "I'll let you have the prize for this race if you run faster than I do, but if you run faster than I do, I'm going to say that running faster than I do is a form of cheating and then I'll ban you from ever competing again." That's just not right.

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  42. I love you. Finally got that nerve racking question answered!! Than you so so much! Love your work, illustrations and audience considerations

  43. The ultimate house edge for casinos has nothing to do with math, it’s psychology. All games have a negative EV with some expected positive fluctuations. When a gambler is ahead but $300, he ‘feels’ great and is willing to risk more and increase his bets because…”it’s the houses money”. That accelerates his looses and guarantees his ruin.

    Thought experiment: if you had a contractual agreement with a casino they would accept your bets for a 24 hour period, no limits. And if you had free use of $10 million dollars. You have zero risk: at the end of the day if you lose the whole 10 mil, no problem, no payback. But you get to keep any profit above 10 mil.

    Perfect setup right? Gambling with someone else’s money lol.

    You then could enter into a low risk martingale game with an initial bet of $1,000. If you lose, the progression is 2k, 4K, 8k etc and you’d have to experience a lot of losses in a row before you lost the whole 10m.

    So you can reasonably expect to play for a little while and bank $1k profit per hand/spin. Now, for most of us 1k is real, nice money. We’d be happy to walk out of any casino with 10k profit.

    You know mathematically that you cannot play forever – at some point there will be 50 losses in a row and the 10m will be long gone. But that probably (haha) won’t happen with the first game you play that day. So it’s safe to play martingale for a little while under these condition (huge bankroll and house agreed rules to not cut you off).

    So you are playing and banking $1k profit every 45 seconds or so. In the wink of an eye you have $10k profit, then $35k. You KNOW that it won’t continue forever. You KNOW that you should stop and go home a winner, 35k is serious money – that’s a new truck or a year of college for one of the kids. But it was only a few easy minutes and winning like this is FUN!

    When to stop?

    Most people push it too far. Most people won’t look down at their chip stack and say ‘damn, that’s $34,690’ and stop right there. They’ll say:

    “Cool, I’ll quit a $35k”
    “If I make it to $50k I’ll cash out”
    “Damn, a win like that will generate a W2G and I need to pay taxes, so that’s really only 22k, I’ll play it up to 75k and pocket 50k after taxes”
    “That only took 20 minutes, so in an hour I should be well over $100k”

    And THAT is the house edge: That voice in your head is what guarantees a win for the casinos.

  44. Interesting tidbit. It's possible to show a profit at blackjack without ever changing your bet. Given a double deck game with 50% penetration, H17, DAS, no surrender, played heads up with an initial investment of $15,000, you can flat bet $25 and wong out at a true count of -1 which will yield a profit of $12 to $13 per 100 hands with a risk of ruin of only 1%. This was calculated using zen count and some of the more basic deviations. The standard deviation is $29.03 per hand. I wouldn't actually recommend playing this, the expectation per hour is actually pretty low given how few hands per hour you'd actually play with all that wonging out.

  45. Coupon is worth nothing. If you win you only get $10 in your example. If you lose you lose nothing. They’ll take the coupon either way

  46. I have a friend who is professional beter (is that one who bets?). He has approximately €200,000 bank and he pores over betting sites on internet almost 8 hours per day. Every time he finds odds that are crossed eg. one gets you 10 to 9 for win and antoher gives you 10 to 9 for lose, he puts half of his bank on the other and half on the other. Last time I talked to him he said that he pulls about €5000 in month. Not much but not bad wage for 8 hours per day.

  47. When he's talking about the positive outcome game-and the result tree-he's talking about being the casino. The longer a player sits-and the smaller the bet-the more likely they are to lose. Which means the more the casino wins.

  48. It's all good but THE SHOE GAME IS DEAD . Most casinos are using continus shuffling machines which is impossible to beat.

  49. If a roulette-wheel has come up red 60 (or even as few as 51) out of the past 100 times, one of two things is possible. First, it's a fair wheel and the predominance of red is just the run of luck it's having. In the long run there will be just as many times that it will come up BLACK 60 times out of the past 100. If the roulette-wheel is fair then red has the same chance of winning as black, and so you can't HURT yourself by betting red if you are determined to bet. The OTHER thing that could be going on is that the wheel has some bias towards numbers that are red. So if you bet red you are betting a color more likely to come up than black. Therefore, you should bet red. If the wheel is fair it's just as good a bet as black, and if the wheel is biased it's biased towards red, so there is no way for black to be a BETTER bet than red although for all we know black MIGHT be EQUALLY likely as red. So bet red.

  50. I'd ay 5$ for such a coupon, any less payment would only profit me, this way I theoretically break even… I think.

  51. If you must gamble, learn to play poker. You can reasonably expect that all casino bets are losers in expectation. Don’t be a loser! Don’t place the bet.

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