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Night of gambling roulette: illusions versa statistics.

Many years ago French mathematicians Blaise Pascal and Pierre de Fermat helped some gamblers to understand chances in their gambling (Wikipedia on-line, 2004). Later Blaise Pascal proposed gambling machine which was a by-product of his perpetual motion devices. This machine was improved several times, banned in various countries, destroyed thousands lives and it still attract people who wants try to beat house. It is roulette, the oldest casino game.

In this assignment we will discuss some theoretical issues of gambling, will spend “night” of gambling and will evaluate our chances to beat the house.

We will use various strategies for gambling. We will use every strategy for one hundred spins only. We will remember that spinning a roulette wheel is an example of deliberate randomization that is similar to random sampling and we will calculate absolute probabilities for this random process. We will remember that random behavior is unpredictable in the short term but predictable in the long term.

Let’s use European (single zero) versions of the roulette wheel (Roulette description, 2004). It has 37 numbers versa 38 on an American (double zero) wheel. If gambler makes bets for even or simple chances than absolute probability of their coming is 0.4865 (18/37) (see table 1 in Appendix). There are different even chances: red and black, even (pair) and odd (impair) and passé (19-36) and manqué (1-18). The probability of these “even” chances is not equal to 0.5 (Hacking, 2001; Gillies, 2000) because of presence of single zero on the wheel. Correspondly in American roulette probability is 18/38=0.4737 only. If I would bet one dollar for any simple chance for 100 spins of the wheel I would double my money on about 48 to 49 bets, but casino has 2.7% advantage in this case. It means that gambler’s chances to win will not come up if he/she will change to red every time black comes up and then change back to black. And if he/she will change to red after there was a run of 4 blacks the absolute probability is still 18/37=0.4865. After four successive blacks, a black is as likely to come as a red. The roulette wheel has no memory!

Straight bet for one number could (zero or 1-36) gives one chance versa thirty-seven so probability is 0.0270 only. Split bet for two numbers (it’s called “a cheval” in French) gives two fold bigger chances 2/36=0,057. Corner (or En Carre) and first four (or Transversale a 4 numeros (0-4)) has probability 4/33= 0,108. Street (or Transversale plein) and combination of zero and two numbers from the first line – 3/34 =0.081 correspondly. Odds for six line Transversale Simple) bet is 6/31=0.162 and for Dozen (Douzaines) (combinations 1-12, 13-24 or 25-36) and columns is 12/25=0.324 only.

Considering one single, isolated spin, the absolute probabilities (Konold, 1991) for straight bet would be the following: p for any number showing = 1, p for a predetermined number showing = 1/37 and p for a predetermined number not showing = 36/37. It means that the probability for a predetermined number during n spins will be equal to 1 – (36/37)n and the probability for a predetermined number not showing during a serie of n spins is (36/37 )n. If we would make 100 bets for the same individual number in order to get a bigger gain than the probability to show the predeterminated number in 100 spins will be 1-(36/37)100 =0.9354. But it does not mean that chances for win will increase with increasing number of roulette spins. Absolute probability will be still 0.0270 only. The solitary probability for a single predetermined number can be obtained by taking the difference between the respective total probabilities SP = ( 36/37 )n-1 – ( 36/37 )n (e.g. for the third spin it’s equal to the total probability for 3rd spins minus total probability for two previous spins). The sequential probability (Konold, 1991) of four individual numbers coming up in a row is very high (1/37 x 1/37 x 1/37 x 1/37 = 1/1,874,161=5,33572E-07), but the absolute probability of predetermined number to come up at each spin is always 1 in 37.

There are several gambling strategies in roulette. Albert Einstein proposed the best. He is reputed to have stated: “You cannot beat a roulette table unless you steal money from it.” (Wikipedia on-line, 2004) But the numerous even money bets in roulette have inspired many players over the years to attempt to beat the game by using one or more variations of a Martingale betting strategy (Wikipedia on-line, 2004). In this strategy the gamer doubles his bet after every loss, so that the first win would recover all previous losses, plus win a profit equal to the original bet. This betting strategy is fundamentally flawed in practice because even bets are not true even and absolute probability for even chances is less than 0.5. Another issue of problem: the size of bet is limited and gambler can not bet infinite sum. But if even roulette have not single zero and casino will loose its advantages than only a gambler with infinite wealth is guaranteed to eventually flip heads. Unfortunately, none of gamblers in fact possessed infinite wealth and the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck. You see, if the first bet will be one dollar and gambler have bad luck in all attempts so in one hudred of bets he will lost 299=633,825,300,114,115,000,000,000,000,000 dollars.

Another popular system is d’Alembert Roulette System or “Pyramide” (Rouleete rules and betting systems, 2004). The “logic” of d’Alembert system is that you will make up more than you lost if you bet more after a loss. Applying this system to even chance bets, gambler increases his/her bet unit on a loss and decrease it on a win. This system does not work because the presence of zero on the wheel. This gives the house 2.7% edge and makes the system flawed. Secondly, the possibility of long runs of the same result make the practicalities of continually increasing gambler’s bet impossible. The D’Alembert is not as potentially damaging as the Martingale but it can still be the cause of very large losses – in one hundred of bets there is damage will be (1+100)*100/2=5050 dollars.

There are many other systems: Labouchere’s or ‘Cancellation’ system, the 1-2-3-4 system, “all but three”, “two for one”, “the lucky 7”, Oscars grid, Thomas Donald’s and Whittacker system etc (Rouleete systems, 2004). The most of them is based on the arithmetical progression and involve outcomes that haven’t hit lately. They focused on the specific numbers, highs or lows, reds or blacks, odds or evens, one or two of the three columns etc. But all these systems use lame logic, pseudo-statistics. Roulette is a game of independent trials (Ellison, 2004). The wheel and ball have no memory. After six or 60 reds in a row, the chance of a red on the next spin is – as always – 18 out of 37. More complicated systems involve detecting irregularities in the apparatus that make some results more likely. If such irregularities (bias) existed, and were great enough, players could indeed gain an edge. Biased wheel systems hold in theory but fail in practice. Roulette wheel construction and maintenance make large biases unlikely, and problems such as broken bearings would be noticed immediately and result in a game being shut down. Small biases require numerous observations and complex calculations. This obstacle limits possibility to use these small bias in the reality. Alan Krigman in his Roulette Strategy & Rules: Can Bias in Roulette Wheels Give You an Edge (1999) provided this example: “The example of the bias on a single number involved a 0.37 percent increase in probability. To be 95 percent confident in detecting this small an anomaly, data would have to be analyzed from roughly 71,000 spins. A 99 percent confidence level would require about 122,500 spins. Neither is even remotely feasible.”

Casino night was tonight. Now we can answer question “Over the course of an evening of gambling at a casino, is it likely that a gambler can beat the house when betting with the game of roulette?”. The answer is NO. We can beat the house only mathematically. In reality casino chances always higher than chances of gamers and roulette systems are not effective. Mathematics and gambling don’t mix. In mathematics we know exactly what is going to happen. Gambling is the exact opposite; you never know what is going to happen – otherwise it wouldn’t be gambling. Statistics help us to understand these laws of successes and failures. Roulette is good example of random processes that could be analyzed by frequentistic statistics.

References:

American Roulette Is Now Mathematically Beatable by R.D. Ellison (2004) at <http://www.thegamblersedge.com/propensity.htm>

Gillies D. (2000) Philosophical theories of probability. Routledge. 223 p.

Hacking I. (2001) An introduction to probability and inductive logic. Cambridge University. 320 p.

Konold, C. (1991). Understanding students’ beliefs about probability. In von Glasersfeld, E. (Ed.), Radical Constructivism in Mathematics Education, Holland, Kluwer, p. 139-156.

Martingale in Wikipedia on-line (September 2004) <http://en.wikipedia.org/wiki/Martingale>

Probability theory of gambling (September 2004) http://www.probability.go.ro

Roulette description (September 2004) <http://www.casino-selection.com/fr/roulette >

Roulette description (September 2004) <http://www.gamblersglen.com/roulette.html>

Roulette in Wikipedia on-line <(September 2004) http://en.wikipedia.org/wiki/Roulette >

Roulette rules (September 2004) http://www.ildado.com/roulette_rules.html

Roulette rules and the betting systems (September 2004) http://www.roulette-student.co.uk

Roulette Strategy & Rules : Can Bias in Roulette Wheels Give You an Edge? by Alan Krigman (1999) in <http://www.acesguidetogambling.com/roulette/wheel_bias.html>

Roulette Strategy (September 2004) <http://onlines-casinos.com/roulette-strategy.shtml>

Roulette strategy (September 2004) <http://www.roulettebucks.com/roulette_strategy.html>

Roulette description (September 2004) http://www.norskcasinoguide.com/rulett.html

Appendix

Table 1 Bets and their absolute probabilities in the roulette.

Bet

Example

Absolute probability

Even chances

Red/black

Even/odd (pair/impair)

Passe/manque

P=18/19=0.4865

Straight

Any number

(0, 1, … , 35, 36)

P=1/36=0.0270

Split

P=2/35=0.054

Corner

P=4/33=0.108

First Four

Street

P=3/34=0.081

Three number including zero

Six-line

P=6/31=0.162

Dozen

1-12 or 13-24 or 25-36

P=12/25=0.324

Column

1st column: 1,4,7,10,13,16,19,22,25,28,31,34

2nd column:

2,5,8,11,14,17,20,23,26,29,32,35

3rd column:

3,6,9,12,15,18,21,24,27,30,33,36

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