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The theoretical win from players who place a wager on the pass line in craps is (0.0141) × (total amount wagered). As a specific example, if players wager $1 per spin on roulette, for one million spins, the house expects to earn about 0.0526 × $1,000,000 = $52,600. Theoretical probability of winning roulette upon deposit on the second day after the first deposit (min £20), and an additional 40 theoretical probability of winning roulette games are given theoretical probability of winning roulette upon deposit on the third day (min £20). Winnings won with games that require deposit, have to be wagered 35x. Bonuses that require deposit, have to be theoretical probability of winning roulette wagered 35x. Karamba.com Welcome Bonus – 100% bonus on your. You know that in European roulette, the probability of a specific number appearing in one spin is 1/37 or 2,7%. But what is the probability of a specific number appearing exactly 1 time in 37 spins? The probability of a specific number appearing exactly 1 time in the course of 37 spins is 0.373 or 37,3%. Mar 19, 2016 Theoretical Win and Expected Value. The average of results is determined by the theoretical win formula. Hence, 50% of all results will achieve this result. For example: A split wager on single-zero roulette pays 17:1. The probability of winning a split bet is 2/37 (2 numbers wagered on from a total of 37 numbers).
- Roulette Odds And Probability
- Theoretical Probability Of Winning Roulette Game
- Roulette Probability Statistics Problem
- Theoretical Probability Of Winning Roulette Numbers
- Probability Of Winning The Lottery
![Probability Probability](/uploads/1/2/5/2/125217512/713180695.jpg)
Theoretical probability of winning roulette upon deposit on the third day (min £20). Winnings won with games that require deposit, have to be wagered 35x. Bonuses that require deposit, have to be theoretical probability of winning roulette wagered 35x. Theoretical Probability of Winning Craps. The game of craps is unique in a couple of ways. For one thing, the game offers some of the best bets in the casino. For another, it also offers some of the worst bets at the same time. Most casino games either have a high house edge or a low house edge; craps has both.
By Ion Saliu, Founder of Roulette Mathematics
This question was posted in mathematical newsgroups (alt.math.recreational, alt.math.undergrad, alt.sci.math.probability): 'Winning and Quitting on Red/Black in Roulette'.
- 'Obviously in roulette betting on in the long run you are going to lose your money but at some point chances are you'll be in profit. To take an extreme example if you had $1000 you could reasonably expect to be up $1 at some point. Is it possible to generalize this? I want to win W dollars at which point I will quit. How much cash C would I need to have probability P of succeeding? Let's say I'm betting on a 37 number roulette wheel (18 red 18 black and one green 0)?'
On the surface, the best probability for the roulette player to be ahead is in one trial (spin): 48.6% to win (versus 51.4% to lose), as far as even-money betting is concerned. I don't agree that it is the best strategy (betting all your bankroll on one spin).
Theoretically, no bankroll will put a player ahead guaranteed, IF flat-betting and playing very long consecutive sessions. There are moments, however, when the roulette player can be ahead by at least one betting unit. Even in even-money bets, the player has a good chance to be ahead by at least one unit after 5, or 10, or even 100 spins. But more than 20 spins are NOT recommended; the probability (odds) to lose go(es) above 50%! Think about it!
The main thing, mathematically, is the number of player's wins in N trials. To be ahead, means the player has won at least one more roulette spin (number of successes) than the number of losses in N trials. The question then becomes:
'What are the probabilities for the player to be ahead in various numbers of trials?'
Everybody can use my probability software SuperFormula, option L: At Least M successes in N trials.
Winning probability: p = 18/37; M must be at least (N/2) + 1.
Here is a number of cases from the player's perspective.
Winning probability: p = 18/37; M must be at least (N/2) + 1.
Here is a number of cases from the player's perspective.
The figures are applicable to all even-money roulette bets: black or red; even or odd; low or high (1-18 or 19-36).
1 trial (spin)
- probability (odds) to win: 48.6%; odds = 1 in 2.05
- probability (odds) to lose: 51.4%; odds = 1 in 1.95
(the probability to lose is 19/37; adding zero to unfavorable cases).
- probability (odds) to win: 48.6%; odds = 1 in 2.05
- probability (odds) to lose: 51.4%; odds = 1 in 1.95
(the probability to lose is 19/37; adding zero to unfavorable cases).
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2 trials (spins)
- probability (odds) to win 2 of 2: 23.7% (1 of 2 doesn't mean 'being ahead')
- probability (odds) to lose 1 of 2: 76.3%
- probability (odds) to win 2 of 2: 23.7% (1 of 2 doesn't mean 'being ahead')
- probability (odds) to lose 1 of 2: 76.3%
3 spins
- probability (odds) to win at least 2 of 3: 48%
- probability (odds) to lose at least 2 of 3: 52%
- probability (odds) to win at least 2 of 3: 48%
- probability (odds) to lose at least 2 of 3: 52%
10 spins
- probability (odds) to win at least 6 of 10: 34.4%; odds = 1 in 2.91
- probability (odds) to lose at least 6 of 10: 41.1%; odds = 1 in 2.43
- probability (odds) to win at least 6 of 10: 34.4%; odds = 1 in 2.91
- probability (odds) to lose at least 6 of 10: 41.1%; odds = 1 in 2.43
20 spins
- probability (odds) to win at least 11 of 20: 36.5%
- probability (odds) to lose at least 11 of 20: 46.2%
- probability (odds) to win at least 11 of 20: 36.5%
- probability (odds) to lose at least 11 of 20: 46.2%
100 spins
- probability (odds) to win at least 51 of 100: 35.5%; odds = 1 in 2.82
- probability (odds) to lose at least 51 of 100: 56.8%; odds = 1 in 1.76.
It's getting worse for the player..
- probability (odds) to win at least 51 of 100: 35.5%; odds = 1 in 2.82
- probability (odds) to lose at least 51 of 100: 56.8%; odds = 1 in 1.76.
It's getting worse for the player..
The roulette strategy (or system) is a totally different ball game! But there are professional gamblers out there, including roulette players! They must have strategies, some roulette systems deduced from some figures like the ones above! The player can be ahead at any point in the game. If so, maybe it's time to move to another (or casino) table: It improves the odds of winning!
Always keep track of the losing and winning streaks. Be strong and put an end to a winning streak. You are ahead, you quit the roulette table. Go to another table and wait until you are ahead. The bankroll is of the essence: It must assure going through long losing streaks. Divide the streaks in 10 spins or 20 spins. Never fight aggressively short or mid-term losing streaks. This is the best approach for those who do not know Ion Saliu's casino gambling systems. A good approach to gambling is the next best thing to a good gambling system! Applicable to blackjack and baccarat, too!
Axiomatic one, everybody knows that the casinos have an edge or house advantage (HA) in all the games they offer, roulette including. The house advantage is created by the payouts in rapport to total possibilities for the respective bet. We can apply this simple formula based on units paid UP over total possibilities TP:
(always expressed as a percentage.)
For example, in single-zero roulette, the one-number (straight-up) bet has payout of 35 to 1. The to qualifier is very important: the casino pays you 35 units and they give you back the unit you bet; thus, you get 36 units. There are 37 possibilities in single-zero roulette: 36 numbers from 1 to 36 plus the 0 number. Therefore, HA = 1 – (UP / TP) = 1 – (36 / 37) = 1 – 0.973 = 0.027 = 2.7%.
Let's calculate HA for the 1 to 1 bets: black/red, even/odd, low/high. HA = 1 – (UP / TP) = 1 – (2 / 2.055) = 1 – 0.973 = 0.027 = 2.7%. There are little differences among bets depending on how many decimal points we work with in our calculations.
The point is, the casinos have an advantage, or the players have a disadvantage. Nonetheless, the players' disadvantage is far better than what they face in state-run lotteries. Yet, most casino gamblers lose big, including at roulette tables. They do not have sufficient bankrolls to withstand long losing streaks.
However, around 45% of the roulette numbers lead the gamblers to profits in a few thousand spins. That is, with a sufficient bankroll, a player has a pretty good chance to make a profit, even if playing a random roulette number, or a favorite number. I analyzed about 8000 roulette spins from Hamburg Spielbank (casino). Quite a few numbers ended up making a profit: roulette systems, magic numbers.
By contrast, the more lottery drawings a player plays, the higher the degree of certainty of a loss. Let's make a comparative analysis to the roulette long series above (spins: total roulette numbers, 37, multiplied by 200). If playing the pick-3 lottery for some 100,000 drawings, it is guaranteed that all pick-3 straight sets will be losers. Some numbers will hit up to 3% to 5% above the norm — but that is not nearly enough to assure a profit. A frequency of 3% to 5% above the norm leads to profits in roulette, however.
Ion Saliu's Paradox and Roulette
Ion Saliu's Paradox of N Trials is presented in detail at saliu.com, especially the probability theory page and the mathematics of gambling formula. If p = 1 / N, we can discover an interesting relation between the degree of certainty DC and the number of trials N. The degree of certainty has a limit, when N tends to infinity. That limit is 1 — 1/e, or approximately 0.632..If you play 1 roulette number for the next 38 spins, common belief was that you expected to win once. Not! Non! Only if you play 38 numbers in 1 spin, your chance to hit the winning number is 100%. Here is an interesting table, which includes also The Free Roulette System #1 presented at the main roulette site.
The maximum gain comes when playing 38 numbers in one spin: 36.3%. Obviously, it makes no sense to play that way because of the house advantage. On the other hand, a so-called wise gambler is more than happy to play one number at a time. What he does is simply losing slowly! Not only that, but losing slowly is accompanied by losing more. That cautious type of gambling is like a placebo. A roulette system such as Free System #1 scares most gamblers. 'Play 34 or 33 numbers in one shot? I'll have a heart attack!' In reality, the Free Roulette System #1 offers a 28.8% advantage over playing singular numbers in long sessions. That's mathematics, and there is no heart to worry about, axiomatic one.
You can also use SuperFormula to calculate all kinds of probabilities and advantage percentages. The option L — At least M successes in N trials is a very useful gambling instrument. If you play 19 numbers in one spin, the probability to win is 50%. If you play 19 numbers in 2 consecutive spins, the probability to win at least once is 75%.
Editor's note
• In an apparent change of heart, the Hamburg casino (Spielbank) offers online roulette results for free. The new link is (for the time being!):
• In an apparent change of heart, the Hamburg casino (Spielbank) offers online roulette results for free. The new link is (for the time being!):
www.spielbank-hamburg.de/spielsaal/permanenzen.php4
Roulette Odds And Probability
• • Real-life roulette spins are also available from the Wiesbaden, Germany, Casino (Spielbank)
www.spielbank-wiesbaden.de/DE/621/Permanenzen2.php: Wiesbaden Spielbank Permanenzen
Roulette: Software, Systems, Super Strategy
See a comprehensive directory of the pages and materials on the subject of roulette, software, systems, and the Super Strategy.- Theory, Mathematics of Roulette Systems, Strategies, Software.
- The Best-Ever Roulette Strategy, Systems based on mathematics of progressions and free-for-all.
- An inventory of free and outrageously pricedroulette systems.
- Software, Systems: Roulette Wheel Positioning, Sectors, Birthday Paradox.
- 'Roulette System that Won Millions!'
~ Ion Saliu's roulette theory, systems, and software were pirated by an Australian gambling group. - James Bond Roulette System for Dozens, Columns.
- Roulette Systems, Threats from Casino Chairman.
- Anti-Gambler Advice from John Patrick, Casino Moleextraordinaire and conspirator.
- Wizard of Odds Had High Praise For Ion Saliu's Gambling Theory.
- Casinos pay troubled individuals to intimidate intelligent gambling system authors.
- Download roulette, blackjack, casino gambling, systems, software.
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Introduction
The following table shows the house edge of most casino games. For games partially of skill perfect play is assumed. See below the table for a definition of the house edge.
Casino Game House Edge
Game | Bet/Rules | House Edge | Standard Deviation |
---|---|---|---|
Baccarat | Banker | 1.06% | 0.93 |
Player | 1.24% | 0.95 | |
Tie | 14.36% | 2.64 | |
Big Six | $1 | 11.11% | 0.99 |
$2 | 16.67% | 1.34 | |
$5 | 22.22% | 2.02 | |
$10 | 18.52% | 2.88 | |
$20 | 22.22% | 3.97 | |
Joker/Logo | 24.07% | 5.35 | |
Bonus Six | No insurance | 10.42% | 5.79 |
With insurance | 23.83% | 6.51 | |
Blackjacka | Liberal Vegas rules | 0.28% | 1.15 |
Caribbean Stud Poker | 5.22% | 2.24 | |
Casino War | Go to war on ties | 2.88% | 1.05 |
Surrender on ties | 3.70% | 0.94 | |
Bet on tie | 18.65% | 8.32 | |
Catch a Wave | 0.50% | d | |
Craps | Pass/Come | 1.41% | 1.00 |
Don't pass/don't come | 1.36% | 0.99 | |
Odds — 4 or 10 | 0.00% | 1.41 | |
Odds — 5 or 9 | 0.00% | 1.22 | |
Odds — 6 or 8 | 0.00% | 1.10 | |
Field (2:1 on 12) | 5.56% | 1.08 | |
Field (3:1 on 12) | 2.78% | 1.14 | |
Any craps | 11.11% | 2.51 | |
Big 6,8 | 9.09% | 1.00 | |
Hard 4,10 | 11.11% | 2.51 | |
Hard 6,8 | 9.09% | 2.87 | |
Place 6,8 | 1.52% | 1.08 | |
Place 5,9 | 4.00% | 1.18 | |
Place 4,10 | 6.67% | 1.32 | |
Place (to lose) 4,10 | 3.03% | 0.69 | |
2, 12, & all hard hops | 13.89% | 5.09 | |
3, 11, & all easy hops | 11.11% | 3.66 | |
Any seven | 16.67% | 1.86 | |
Double Down Stud | 2.67% | 2.97 | |
Heads Up Hold 'Em | Blind pay table #1 (500-50-10-8-5) | 2.36% | 4.56 |
Keno | 25%-29% | 1.30-46.04 | |
Let it Ride | 3.51% | 5.17 | |
Pai Gowc | 1.50% | 0.75 | |
Pai Gow Pokerc | 1.46% | 0.75 | |
Pick ’em Poker | 0% - 10% | 3.87 | |
Red Dog | Six decks | 2.80% | 1.60 |
Roulette | Single Zero | 2.70% | e |
Double Zero | 5.26% | e | |
Sic-Bo | 2.78%-33.33% | e | |
Slot Machines | 2%-15%f | 8.74g | |
Spanish 21 | Dealer hits soft 17 | 0.76% | d |
Dealer stands on soft 17 | 0.40% | d | |
Super Fun 21 | 0.94% | d | |
Three Card Poker | Pairplus | 7.28% | 2.85 |
Ante & play | 3.37% | 1.64 | |
Video Poker | Jacks or Better (Full Pay) | 0.46% | 4.42 |
Wild Hold ’em Fold ’em | 6.86% | d |
Notes
a | Liberal Vegas Strip rules: Dealer stands on soft 17, player may double on any two cards, player may double after splitting, resplit aces, late surrender. |
b | Las Vegas single deck rules are dealer hits on soft 17, player may double on any two cards, player may not double after splitting, one card to split aces, no surrender. |
c | Assuming player plays the house way, playing one on one against dealer, and half of bets made are as banker. |
d | Yet to be determined. |
e | Standard deviation depends on bet made. |
f | Slot machine range is based on available returns from a major manufacturer |
g | Slot machine standard deviation based on just one machine. While this can vary, the standard deviation on slot machines are very high. |
House Edge
The house edge is defined as the ratio of the average loss to the initial bet. The house edge is not the ratio of money lost to total money wagered. In some games the beginning wager is not necessarily the ending wager. For example in blackjack, let it ride, and Caribbean stud poker, the player may increase their bet when the odds favor doing so. In these cases the additional money wagered is not figured into the denominator for the purpose of determining the house edge, thus increasing the measure of risk.
The reason that the house edge is relative to the original wager, not the average wager, is that it makes it easier for the player to estimate how much they will lose. For example if a player knows the house edge in blackjack is 0.6% he can assume that for every $10 wager original wager he makes he will lose 6 cents on the average. Most players are not going to know how much their average wager will be in games like blackjack relative to the original wager, thus any statistic based on the average wager would be difficult to apply to real life questions.
The conventional definition can be helpful for players determine how much it will cost them to play, given the information they already know. However the statistic is very biased as a measure of risk. In Caribbean stud poker, for example, the house edge is 5.22%, which is close to that of double zero roulette at 5.26%. Free triple diamond slots. However the ratio of average money lost to average money wagered in Caribbean stud is only 2.56%. The player only looking at the house edge may be indifferent between roulette and Caribbean stud poker, based only the house edge. If one wants to compare one game against another I believe it is better to look at the ratio of money lost to money wagered, which would show Caribbean stud poker to be a much better gamble than roulette.
Many other sources do not count ties in the house edge calculation, especially for the Don’t Pass bet in craps and the banker and player bets in baccarat. The rationale is that if a bet isn’t resolved then it should be ignored. I personally opt to include ties although I respect the other definition.
Allen-Bradley uses the capital letter “I” to designate a hardwired input. An address that describes an input on an SLC 500 is I:4/0. Similar to the output structure, I:4/0 means that it is a physical input. I:4/0 means that it uses Slot 4 (the 5th slot in the rack). I:4/0 means that it is the first input on the card.
Within each of the Allen-Bradley PLC’s “files” are multiple “elements,” each element consisting of a set of bits (8, 16, 24, or 32) representing data. Elements are addressed by number following the colon after the file designator, and individual bits within each element addressed by a number following a slash mark.
![Plc](http://engineeronadisk.com/notes_mechtron/images/functiona.gif)
Element of Risk
Roulette Probability Statistics Problem
For purposes of comparing one game to another I would like to propose a different measurement of risk, which I call the 'element of risk.' This measurement is defined as the average loss divided by total money bet. For bets in which the initial bet is always the final bet there would be no difference between this statistic and the house edge. Bets in which there is a difference are listed below.
Theoretical Probability Of Winning Roulette Numbers
Element of Risk
Game | Bet | House Edge | Element of Risk |
---|---|---|---|
Blackjack | Atlantic City rules | 0.43% | 0.38% |
Bonus 6 | No insurance | 10.42% | 5.41% |
Bonus 6 | With insurance | 23.83% | 6.42% |
Caribbean Stud Poker | 5.22% | 2.56% | |
Casino War | Go to war on ties | 2.88% | 2.68% |
Heads Up Hold 'Em | Pay Table #1 (500-50-10-8-5) | 2.36% | 0.64% |
Double Down Stud | 2.67% | 2.13% | |
Let it Ride | 3.51% | 2.85% | |
Spanish 21 | Dealer hits soft 17 | 0.76% | 0.65% |
Spanish 21 | Dealer stands on soft 17 | 0.40% | 0.30% |
Three Card Poker | Ante & play | 3.37% | 2.01% |
Wild Hold ’em Fold ’em | 6.86% | 3.23% |
Standard Deviation
The standard deviation is a measure of how volatile your bankroll will be playing a given game. This statistic is commonly used to calculate the probability that the end result of a session of a defined number of bets will be within certain bounds.
The standard deviation of the final result over n bets is the product of the standard deviation for one bet (see table) and the square root of the number of initial bets made in the session. This assumes that all bets made are of equal size. The probability that the session outcome will be within one standard deviation is 68.26%. The probability that the session outcome will be within two standard deviations is 95.46%. The probability that the session outcome will be within three standard deviations is 99.74%. The following table shows the probability that a session outcome will come within various numbers of standard deviations.
I realize that this explanation may not make much sense to someone who is not well versed in the basics of statistics. If this is the case I would recommend enriching yourself with a good introductory statistics book.
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Standard Deviation
Number | Probability |
---|---|
0.25 | 0.1974 |
0.50 | 0.3830 |
0.75 | 0.5468 |
1.00 | 0.6826 |
1.25 | 0.7888 |
1.50 | 0.8664 |
1.75 | 0.9198 |
2.00 | 0.9546 |
2.25 | 0.9756 |
2.50 | 0.9876 |
2.75 | 0.9940 |
3.00 | 0.9974 |
3.25 | 0.9988 |
3.50 | 0.9996 |
3.75 | 0.9998 |
Hold
Although I do not mention hold percentages on my site the term is worth defining because it comes up a lot. The hold percentage is the ratio of chips the casino keeps to the total chips sold. This is generally measured over an entire shift. For example if blackjack table x takes in $1000 in the drop box and of the $1000 in chips sold the table keeps $300 of them (players walked away with the other $700) then the game's hold is 30%. If every player loses their entire purchase of chips then the hold will be 100%. It is possible for the hold to exceed 100% if players carry to the table chips purchased at another table. A mathematician alone can not determine the hold because it depends on how long the player will sit at the table and the same money circulates back and forth. There is a lot of confusion between the house edge and hold, especially among casino personnel.
Hands per Hour, House Edge for Comp Purposes
The following table shows the average hands per hour and the house edge for comp purposes various games. The house edge figures are higher than those above, because the above figures assume optimal strategy, and those below reflect player errors and average type of bet made. This table was given to me anonymously by an executive with a major Strip casino and is used for rating players.
Hands per Hour and Average House Edge
Games | Hands/Hour | House Edge |
---|---|---|
Baccarat | 72 | 1.2% |
Blackjack | 70 | 0.75% |
Big Six | 10 | 15.53% |
Craps | 48 | 1.58% |
Car. Stud | 50 | 1.46% |
Let It Ride | 52 | 2.4% |
Mini-Baccarat | 72 | 1.2% |
Midi-Baccarat | 72 | 1.2% |
Pai Gow | 30 | 1.65% |
Pai Pow Poker | 34 | 1.96% |
Roulette | 38 | 5.26% |
Single 0 Roulette | 35 | 2.59% |
Casino War | 65 | 2.87% |
Spanish 21 | 75 | 2.2% |
Sic Bo | 45 | 8% |
3 Way Action | 70 | 2.2% |
Probability Of Winning The Lottery
Translation
A Spanish translation of this page is available at www.eldropbox.com.
Written by: Michael Shackleford