The second gamblers falacy (mathy)
December 9th, 2013 at 4:31:36 PM
permalink
Recently took a trip to my first casino, I was playing 100x odds $10 table I walked away with +$7000 after placing about 50 don't come bets (woohoo!).
Later this got us into a nerdy math discussion. I thought it would be interesting to share.
I proposed that low EV games like non-counting blackjack do not have a better "player advantage" compared to games such as single # on roulette or even keno. and the idea that because a game returns less as a percent bet is not the correct approach to finding the casino game with the best "player advantage".
Let's take two games: European roulette, and 2x odds craps.
Playing $1 on a single number in European roulette has a return of $0.027 and a standard deviation of $5.84
Playing $2 come bet with 2x odds has a return of -$0.027 and a standard deviation of $5.71
If i bet these games at the same frequency the outcome will be virtually identical.
After say 300 plays (that is enough so a normal distribution will approximate the roulette game):
roulette: EV -$8.11, sigma $101.11
craps: EV -8.48, sigma $99.00
You might say the velocity of reaching that point is different. But, you could match the velocity AND the final distribution of outcomes in either game by adjusting the frequency of bets and wager of the other.
So what determines what casino game has the best "player advantage". I'd say its the ratio of the standard deviation to the EV. not just the EV.
Later this got us into a nerdy math discussion. I thought it would be interesting to share.
I proposed that low EV games like non-counting blackjack do not have a better "player advantage" compared to games such as single # on roulette or even keno. and the idea that because a game returns less as a percent bet is not the correct approach to finding the casino game with the best "player advantage".
Let's take two games: European roulette, and 2x odds craps.
Playing $1 on a single number in European roulette has a return of $0.027 and a standard deviation of $5.84
Playing $2 come bet with 2x odds has a return of -$0.027 and a standard deviation of $5.71
If i bet these games at the same frequency the outcome will be virtually identical.
After say 300 plays (that is enough so a normal distribution will approximate the roulette game):
roulette: EV -$8.11, sigma $101.11
craps: EV -8.48, sigma $99.00
You might say the velocity of reaching that point is different. But, you could match the velocity AND the final distribution of outcomes in either game by adjusting the frequency of bets and wager of the other.
So what determines what casino game has the best "player advantage". I'd say its the ratio of the standard deviation to the EV. not just the EV.
December 9th, 2013 at 4:37:28 PM
permalink
This topic has been discussed before. I fully recognize that most recreational gamblers like variance. If there is some statistic that factors in both mean and variance I'm happy to talk about it.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 9th, 2013 at 4:43:09 PM
permalink
The Sharpe ratio?Quote: WizardThis topic has been discussed before. I fully recognize that most recreational gamblers like variance. If there is some statistic that factors in both mean and variance I'm happy to talk about it.
Climate Casino: https://climatecasino.net/climate-casino/
December 9th, 2013 at 5:09:40 PM
permalink
Quote: teliotThe Sharpe ratio?
For the benefit of others, the Sharp Ratio would be the house edge divided by standard variance, in the context of a casino game. Correct me if I'm wrong.
That would be useful for advantage players who shun variance. However, they already have the Kelly ratio, which is advantage/variance. I'm looking for a metric useful for recreation gamblers who want variance. Perhaps house edge * standard deviation. However, I'd prefer to use a known statistic, as opposed to pulling something out of my wazoo.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 9th, 2013 at 6:13:39 PM
permalink
Nice first post
I say we need about 1500 rounds for Roulette to be looking normal and get away from the long tail to the right
$5 Roulette bets
If not,
is fair to compare a bet over 300 rounds of play at $300 total handle (roulette)
to a bet with over $1483 total handle (way higher avg bet per round)?
and one is impressed that the ev and sd is about the same but one player wagered almost 5 times more actual $$$?
Craps player with Full 2X odds ($5 odds on 6&8) has an average bet of $4.944
we call that $5
EV: -$8.484848485
SD: $102.35
ev/sd: -0.082897367 (close to 0 is good)
Now Roulette can bet $5 for 297 rounds
1485 vs 1483 action
now that looks fair to me
Roulette
EV: -$40.13513514 (sure looks higher)
$503.0373777 (now this is good)
ev/sd = -0.079785592
Now what to compare?
Was this fair?
Quote: chrisrIf i bet these games at the same frequency the outcome will be virtually identical.
After say 300 plays (that is enough so a normal distribution will approximate the roulette game):
I say we need about 1500 rounds for Roulette to be looking normal and get away from the long tail to the right
$5 Roulette bets
How about comparing total action (handle) in the mix or even average wager.Quote: chrisrroulette: EV -$8.11, sigma $101.11
craps: EV -8.48, sigma $99.00
You might say the velocity of reaching that point is different. But, you could match the velocity AND the final distribution of outcomes in either game by adjusting the frequency of bets and wager of the other.
If not,
is fair to compare a bet over 300 rounds of play at $300 total handle (roulette)
to a bet with over $1483 total handle (way higher avg bet per round)?
and one is impressed that the ev and sd is about the same but one player wagered almost 5 times more actual $$$?
Craps player with Full 2X odds ($5 odds on 6&8) has an average bet of $4.944
we call that $5
EV: -$8.484848485
SD: $102.35
ev/sd: -0.082897367 (close to 0 is good)
Now Roulette can bet $5 for 297 rounds
1485 vs 1483 action
now that looks fair to me
Roulette
EV: -$40.13513514 (sure looks higher)
$503.0373777 (now this is good)
ev/sd = -0.079785592
Now what to compare?
Was this fair?
winsome johnny (not Win some johnny)
December 9th, 2013 at 6:13:40 PM
permalink
@wiz
I've heard the term relative standard deviation used. RSD=|(standard deviation/average)|
If you want the most variance you should just place a single large bet. since the RSD for dividing your wager over repeated plays is sqrt(n)/n*RSD.
i think a useful metric would be the chance of winning a game at 1,000.. 10,000.. 100,000 plays for various RSD
@johnny
my point was that given the same RSD a low-stakes player of high edge game is going to end up in the same distribution of outcomes as a higher-stakes player of a low edge game (most of the time)... and the metric that really matters for beating the house is RSD.
--
the other game i played for $5 a hand was let it ride, another $5/hand player scoffed at another for playing the $1 side bet.
$1 side bet:~$30/-$0.20 $5 main bet: ~$25/-$0.175
the distribution of the two bets again are nearly identical (for large N, slightly different skewness for small n)
..just a humorous thing about gamblers who look down on those penny slot players etc., they end up in the exact same distribution as them (by the ingenious design of modern casinos).
I've heard the term relative standard deviation used. RSD=|(standard deviation/average)|
If you want the most variance you should just place a single large bet. since the RSD for dividing your wager over repeated plays is sqrt(n)/n*RSD.
i think a useful metric would be the chance of winning a game at 1,000.. 10,000.. 100,000 plays for various RSD
@johnny
my point was that given the same RSD a low-stakes player of high edge game is going to end up in the same distribution of outcomes as a higher-stakes player of a low edge game (most of the time)... and the metric that really matters for beating the house is RSD.
--
the other game i played for $5 a hand was let it ride, another $5/hand player scoffed at another for playing the $1 side bet.
$1 side bet:~$30/-$0.20 $5 main bet: ~$25/-$0.175
the distribution of the two bets again are nearly identical (for large N, slightly different skewness for small n)
..just a humorous thing about gamblers who look down on those penny slot players etc., they end up in the exact same distribution as them (by the ingenious design of modern casinos).
December 9th, 2013 at 6:28:17 PM
permalink
deleted
DUHHIIIIIIIII HEARD THAT!
December 9th, 2013 at 7:22:04 PM
permalink
Quote: IbeatyouracesRecreational gamblers do not care about any of this. .
Correct. Why anybody would care about it is a
mystery. Unless you have a bet that beats the
HE, all this info is useless.
"It's not called gambling if the math is on your side."
December 9th, 2013 at 7:27:30 PM
permalink
Always being negative. Surely one of these systems has to work. I mean I see people winning everyday.
Shed not for her
the bitter tear
Nor give the heart
to vain regret
Tis but the casket
that lies here,
The gem that filled it
Sparkles yet
December 9th, 2013 at 7:33:19 PM
permalink
Quote: EvenBobCorrect. Why anybody would care about it is a
mystery. Unless you have a bet that beats the
HE, all this info is useless.
ya, it's useless I'll admit that. i just wanted to educate the optimal strategy craps players not to look down on their brethren penny slot players.
December 9th, 2013 at 7:54:47 PM
permalink
Do you see people losing everyday also?Quote: BuzzardAlways being negative. Surely one of these systems has to work. I mean I see people winning everyday.
Some points to consider.
Original Post: Roulette ... I think there is a "minus sign" missing at that 0.27 stuff.
Later Posts about Actual Behavior versus Mathematics is always valid but also of great interest.
We bet on "lucky" numbers, birthdates, etc. We "feel" the next number will be Red. We might be doing some mental Sharpe Ratio calculations but that woman's perfume changes the numbers just about as much as that last free drink we had changed a few numbers. And if we are in a group we have various social factors to deal with so if we ourselves are sober but the "group" wants to play The Big Six Wheel then there is liable to be a bunch of drunks playing the Big Six Wheel one of whom is also doing Sharpe Ratios.
If this is our weekly trip to a casino the behavior will be different than if it is a once a year trip. The math won't change but the behavior will.
Casinos know the women bet on colors, horses names, city names, etc. And they often win that way. And they often lose that way. Its called fun.
Its nice to know the math, just as it may be nice to know the chemistry behind the effect that woman's perfume is having on you ... but it doesn't necessarily change the behavior.
December 9th, 2013 at 7:54:47 PM
permalink
Duplicate deleted. (Keyboard's fault).
December 9th, 2013 at 8:03:24 PM
permalink
Or alcohol, could be either.
Shed not for her
the bitter tear
Nor give the heart
to vain regret
Tis but the casket
that lies here,
The gem that filled it
Sparkles yet
December 10th, 2013 at 7:53:50 AM
permalink
I think my op was too verbose to get my point across, I like my let it ride example better..
the $1 "sucker bet" has a u=-.22 sigma=29.0
the $5 main bet has u=-.18 sigma=27.8
You are making nearly the same bet with either choice.. which is very interesting since the psychology is that the former is a sucker bet..
the $1 "sucker bet" has a u=-.22 sigma=29.0
the $5 main bet has u=-.18 sigma=27.8
You are making nearly the same bet with either choice.. which is very interesting since the psychology is that the former is a sucker bet..
December 10th, 2013 at 6:01:53 PM
permalink
Quote: 7crapsev/sd
As something to go by, this always made sense to me, since the longer you play, the more the EV approaches one SD and total foolishness ... as it takes one standard deviation of good luck just to break even, if the EV hits one SD.
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: “Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell!” She is, after all, stone deaf. ... Arnold Snyder
December 10th, 2013 at 10:08:08 PM
permalink
There really is no one stat that will make everyone happy in determining what is a good bet.Quote: odiousgambit... since the longer you play, the more the EV approaches one SD and total foolishness ... as it takes one standard deviation of good luck just to break even, if the EV hits one SD.
EV has been used for a very long time.
the ev/sd is not a very good indicator when a bet has non-zero skewness
especially high odds payoff type bets (think video poker)
The OP example and the one I did at a $5 bet
showed a slightly smaller ev/sd value for Roulette but over 300 bets
Roulette has a 57.7% chance of showing a net loss
Craps has a 54.6% chance of showing a net loss
Craps
EV: -$8.48
SD: $102.35
ev/sd: -0.082897367
roulette
EV: -$8.11
SD: $101.11
ev/sd: -0.080209673
Craps Pass Line $2 with Full 2X odds
Roulette $1 straight-up
the skewness shows in the graph
Craps is just about normal (green curve)
both bets may look close by ev and sd but the graph shows they can be quite different at a small number of trials.
different strokes for different folks
here is what 2 bets that have the same distribution looks like
Pass with 10x odds and DPass with 10X Lay odds
look close to see the two different colors of each curve
winsome johnny (not Win some johnny)
December 10th, 2013 at 11:16:58 PM
permalink
In your first graph, shouldn't the areas under the graphs look about the same... they don't seem to be even close...
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
December 10th, 2013 at 11:18:17 PM
permalink
Quote: chrisrya, it's useless I'll admit that. i just wanted to educate the optimal strategy craps players not to look down on their brethren penny slot players.
Sneeches. Lol...
If the House lost every hand, they wouldn't deal the game.
December 11th, 2013 at 12:07:00 AM
permalink
going by just ev and sd one would think so.Quote: thecesspitIn your first graph, shouldn't the areas under the graphs look about the same... they don't seem to be even close...
even at 3000 bets we have the same curves but the Roulette bet is now looking very normal
here is a better view of the Craps Pass with Full 2X odds. remember we have 6 possible outcomes
and only 2 with the Roulette bet.
Too late for me to calculate the skew and kurtosis for the bets right now.
notice the y-axis
even Excel shows the same
winsome johnny (not Win some johnny)
December 11th, 2013 at 12:25:15 AM
permalink
Thanks for the fancy graphics.
I suspect a lot of thought and work goes into them even if I haven't the foggiest as to what is being depicted.
I suspect a lot of thought and work goes into them even if I haven't the foggiest as to what is being depicted.
December 11th, 2013 at 10:31:54 AM
permalink
I think many Recreational Players want a game with Low House Edge but High Variance and the Variance Skewed towards Positive Results. Narrow and Fat Tail for Negative results and Wider and Thiner Tail for Positive results.
ie Lose more often but smaller amounts and win less often but Higher amounts.
That's why the enjoy games where the Payouts are more than 1:1 and there can be payouts like 10:1 and more.
Recreational Players get there enjoyment from these High Win sessions. So a 'good' game should result in the occasional High Win session which should be much Higher than their usual low losing session.
BUT for regular recreational players when the HE is too high for a specific game, they eventually get the feel of it and move to another game.
I do not think it is easy to make a metric to calculate this.
ie Lose more often but smaller amounts and win less often but Higher amounts.
That's why the enjoy games where the Payouts are more than 1:1 and there can be payouts like 10:1 and more.
Recreational Players get there enjoyment from these High Win sessions. So a 'good' game should result in the occasional High Win session which should be much Higher than their usual low losing session.
BUT for regular recreational players when the HE is too high for a specific game, they eventually get the feel of it and move to another game.
I do not think it is easy to make a metric to calculate this.
December 11th, 2013 at 11:45:40 AM
permalink
im not sure skewness really makes a huge difference.. for payouts such as 10:1 it won't take very many repeated plays before you can approximate the results with a normal distribution..
games with 1,000:1 payouts will still show some skewness after a session of plays, but regardless i think standard deviation is still a good metric for perceived volatility.
games with 1,000:1 payouts will still show some skewness after a session of plays, but regardless i think standard deviation is still a good metric for perceived volatility.
December 11th, 2013 at 12:44:18 PM
permalink
I have barely ever played roulette. My experience is limited individual even payout bets on my way into or out of casino. My question is: Are you allowed to play $1 on a single number at a $5 (or more) roulette table?
December 12th, 2013 at 4:30:14 AM
permalink
Quote: endermikeI have barely ever played roulette. My experience is limited individual even payout bets on my way into or out of casino. My question is: Are you allowed to play $1 on a single number at a $5 (or more) roulette table?
Yes, as long as your total inside bets are at least table minimum. For example, $1 on five different inside bets totaling $5 works.
December 12th, 2013 at 9:31:00 AM
permalink
In Excel it takes less than a minute of work after the initial code is complete.Quote: FleaStiffThanks for the fancy graphics.
I suspect a lot of thought and work goes into them even if I haven't the foggiest as to what is being depicted.
The other program does all this graph on the fly after selecting the bet once you enter it into the program.
It is saved for future use.
The graph is showing the net results from 300, $2 pass line wagers and Full 2X odds
($4 odds outside #s and $5 odds on the 6&8)
Six possible bankroll moves per pass line decision
(each multiplied and added at each round to the previous round results - the 1 in 100 million probabilities and higher are dropped off)
-$2 come out craps roll
-$6 outside point loss
-$7 6or8 point loss
$2 pass come out win
$8 inside point win
$10 4 or 10 point win
Here is a photo of the first 5 decisions, then 10 then 100
at 5 we can see the Bell curve taking shape and it really is seen even at 10 rounds completed (second to last graph).
winsome johnny (not Win some johnny)
December 12th, 2013 at 10:32:20 AM
permalink
Quote: 7crapsHere is a photo of the first 5 decisions, then 10 then 100
at 5 we can see the Bell curve taking shape and it really is seen even at 10 rounds completed (second to last graph).
the CLT in action :)
December 12th, 2013 at 11:32:56 AM
permalink
yes, the good old central limit theoremQuote: chrisrthe CLT in action :)
given enough trials, most distributions (from casino bets) arrive at that normal curve, some earlier than others.
again the ev and sd for
the $2 craps pass line with full 2x odds
-$0.028282828
$5.91
$1 0Roulette straight up bet
-$0.027027027
$5.837837838
they sure look very close. and over time they should be still close.
But more time is required than what most think.
The graphs show how different the paths are as the trials increase to being normal.
The Roulette has lost almost all skew at 3k trials
1-5, 10, 100, 3k spins
the 3k graph shows 100 possible outcomes with a probability equal to or greater than 0.000001% (rounded)
winsome johnny (not Win some johnny)
December 12th, 2013 at 11:40:37 AM
permalink
Do $1 tables exist anywhere you want go?
The reason I ask is because where I tend to go (vegas and NA casinos) I commonly see $5 roulette with double zero. Sometimes (off hours generally) I can find $5 craps, and always $10. I also have no trouble finding games with at least 3-4-5 odds. What is the lowest limit people see single zero wheels and where?
My point being that I'm guessing the OP argument falls apart when we consider the games prevalent most places.
The reason I ask is because where I tend to go (vegas and NA casinos) I commonly see $5 roulette with double zero. Sometimes (off hours generally) I can find $5 craps, and always $10. I also have no trouble finding games with at least 3-4-5 odds. What is the lowest limit people see single zero wheels and where?
My point being that I'm guessing the OP argument falls apart when we consider the games prevalent most places.
December 12th, 2013 at 11:45:25 AM
permalink
deleted
DUHHIIIIIIIII HEARD THAT!
December 12th, 2013 at 11:47:08 AM
permalink
Regarding the question about term for mean vs variance. In the journals I frequent (I'm an applied math/stat PhD student) "coefficient of variation" is the preferred measure:
http://en.wikipedia.org/wiki/Coefficient_of_variation
It is the ratio of sigma to mu. It has it's disadvantages, but for my money it is the best "single number summary" of what we are talking about.
http://en.wikipedia.org/wiki/Coefficient_of_variation
It is the ratio of sigma to mu. It has it's disadvantages, but for my money it is the best "single number summary" of what we are talking about.
December 12th, 2013 at 11:55:28 AM
permalink
He was just using the $1 Roulette bet as an example showingQuote: endermikeMy point being that I'm guessing the OP argument falls apart when we consider the games prevalent most places.
a reason not to use EV (expected value) only when determining what is a good bet.
I would agree with you that $1 straight up bets total inside bets are long gone at most US casinos.
25cent and 10cent games were popular in the 1990s in Reno. I dealt them.
So the OP could change his Roulette bet to $5
for an ev and sd of
-$0.135135135
$29.18918919
and the Craps bet to $10 flat full 2X odds
-$0.141414141
$29.55
winsome johnny (not Win some johnny)
December 12th, 2013 at 12:09:23 PM
permalink
Again, I refer to my last posts. The $5 roulette will be double zero. If we are willing to gamble at those levels in the US, at that point it will be easy to find a craps game where 5x or 10x odds will beat the wheel solidly.
I agree with the thought behind OP's argument that mean without speed (bets per time) and variance lacks context, but let's not act like I could find a $1 craps table with 100x odds (the equivalent of OP's post). Although Casino Royale used to (and may still) have a game within shouting distance of that.
I agree with the thought behind OP's argument that mean without speed (bets per time) and variance lacks context, but let's not act like I could find a $1 craps table with 100x odds (the equivalent of OP's post). Although Casino Royale used to (and may still) have a game within shouting distance of that.
December 12th, 2013 at 1:33:23 PM
permalink
Quote: endermikeAgain, I refer to my last posts. The $5 roulette will be double zero. If we are willing to gamble at those levels in the US, at that point it will be easy to find a craps game where 5x or 10x odds will beat the wheel solidly.
I agree with the thought behind OP's argument that mean without speed (bets per time) and variance lacks context, but let's not act like I could find a $1 craps table with 100x odds (the equivalent of OP's post). Although Casino Royale used to (and may still) have a game within shouting distance of that.
I think actually at my local casino they let you play single $1 bets in roulette, at least they let people stand around a crowded craps table and play the $1 center bets without placing a table min bet..
in any case, there are plenty of practical examples, the favorite one I've seen so far (i posted earlier)
$1 side bet in let it ride versus $5 main bet. i.e a $5+1 player would more or less be on the same footing (equivalent mu/sigma) as a $10 player.. and the $10 player might think the $5+1 player is a sucker for placing the side bet.
there are probably quite a few $1 side bets that would follow the same logic.
