## Standard American Roulette Wheel

• An American roulette wheel has 38 equal-sized divisions. First there are the numbers 1 through 36. These are joined by a pair of zeros – featuring the numbers ‘0’ and ‘00’. In other versions of the game, such as European and French roulette wheels, you’ll find just a single zero.
• The standard American roulette wheel consist of 38 slots, numbered 0, 00 and 1 through 36. The slots numbered 0 and 00 are green, of the remaining slots, exactly half are Red and half are black. On each ply, a ball is bounced on the turning wheel and it land randomly in one of the slots. Players are allowed to bet on specific numbers, specific colors, or in other words. Suppose that you walk.
• I've seen quoted in a number of places that the standard deviation for a 1-number bet on a 38 number Roulette wheel (0, 00, 1, 2,., 36) is 5.76. I can't seem to find a single source that shows how this calculation is made.

American Roulette. The free American roulette is largely similar to the European one. However, the main difference between the American roulette wheel and that of the European version is the extra zero: the double zero (00) pocket, which brings about a higher house edge for the former.

Roulette is among the oldest casino games still in operation. People variously attribute the invention of the game to Blaise Pascal, Italian mathematician Don Pasquale and some others. Anyway, the introduction of the roulette wheel in Paris happened in the year 1765.

The American roulette wheel boasts 38 slots, which feature numbers 00, 0 and 1-36. The slots 0,00 are green while slots 1,3,5,7,9,112,14,16,18,19,21,23,25,27,30,32,34, and 36 are red.

Slots 2,4,6,8,10,11,13,15,17,20,22,24,26,28,29,31,33, and 35 are black. Apart from the 0 and 00 slots, all the other slots alternate between black and red. The designers had one goal when arranging the numbers in a strange way. Their goal was to ensure that the high and low numbers, as well as the even and odd numbers, appear to alternate.

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Roulette experiment is quite simple. The gambler spins the wheel and then rolls a small ball in a groove, in opposite direction as the wheel’s motion. Eventually, the ball drops into a slot. Naturally, gamblers assume that the roulette wheel is fair. Therefore, the random variable X that provides the ball’s slot number remains evenly distributed over the sample.

S={00,0,1,…,36}

Space S={00,0,1,…,36}, therefore,
1
P(X=x)=38 x∈S
P(X=x)=138 for each x∈S

The bets

When it comes to gambling, roulette is a very popular game due to its wider variety of bets the gambler can make. Most bets will appear to have similar expected values – negative value. But the variance will highly differ depending on the particular bet.

All roulette bets share the same expected value. However, the standard deviation will vary inversely with the total number of numbers the gambler selects. What are the significances of that for the gambler?

The theoretical win, the expected value and the standard deviation

The gambler derives the theoretical win form possibilities built into the casino game. Theoretically, all casino games are designed in a certain way to guarantee a good return to the casino, the winnings for the players (the theoretical wins) are always negative while the expectation also known as the expected value (winnings for the particular casino) is usually positive.

But as you expect with any probability, a randomness element exists. Therefore, the theoretical win standard deviation offers a threshold for the casino managers to decide whether the play has passed a limit where it turns suspect.

Let it ride game rules. The idea of the game is to claim train routes across turn-of-the-century Europe. You start off with some 'tickets' which describe the cities you need to join with train routes. You claim routes by collecting sets of cards, so fundamentally Ticket to Ride is a set collecting game in the rummy family. Our board game index is loaded with all your favorite classic board games and most likely ones you have never heard of before. To get the most fun out of your games, you need to understand the game rules. Don't let the other team change or even make up their own rules or instructions in their favor. Let it Ride is a simple poker-based game based on the poker value of a final five-card hand. Start by making three equal bets. This is done by clicking a chip and then clicking on the spot marked with a \$ sign on the table. The game will put an equal bet on the other two spots for you.

Theoretical win and the expected value

Gamblers determine the average results using theoretical win formula. A quick example:

The payout of a split bet on single-zero roulette is 17:1. And 2/17 is the possibility of winning the split bet. In other words, 2 numbers are played from a total of 37 numbers. The possibility of losing the wager is 37-2/37=35/37 – for other numbers not played on. The wager theoretical winning formula is:
(Possibility of winning x units of wager payout) – (possibility of losing x bet)

(2/37 x17) – (35/37 x 1) = -0.02703

Alternatively, 50% of every wager on the single-zero roulette split wagers, in the long term; will achieve a \$0.02703 loss per every dollar the gambler bets.
The standard deviation

Mostly, results will fall in the range of 3-standard deviations from the mean. To obtain the standard deviation, you have to turn to binomial distribution. The binomial distribution is a statistical dispersion where only 1 of every 2 outcomes is possible. In the above case, the outcome is either a loss or a win.

The binomial standard deviation formula is:

Square Root (the total number of games x the possibility of winning x the possibility of losing) x the average wager x (the unit of wager + the wager units’ payout)
Using the split wager example and \$2 over 3 games as the average, the standard deviation formula would be:
Square Root (3 x 2/37 x 35/37 x \$2 x (1+17)
= Square Root (3 x 2/37) x \$2 x 18
= \$14.09972 (Standard deviation of the theoretical win)
The result
Gamblers will use Z-table and the following table to determine whether the win of a gambler is probable.
(Win – Average win (the theoretical win)) / Standard deviation
If the player in the above example won \$15, the yield would be:
The gambler wagered 3 games with a mean wager of \$2. Therefore, the total bet would be \$6
(15 – (-0.02703 x6)) / 14.09972
= 1.075352

The result of the wager is 1.075352, which means that the player’s gaming result is 1.075352-standard deviation from the mean. In other words, the gaming results of the player are in the range of the results 15%.
Importance of standard deviation in roulette casino

Actually, some wagers are more probable than the others are. Therefore, as a wager you will need to look at the probabilities. Probability is a basic ways of measuring the possibility of something happening. It is important in roulette casino game.

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First time roulette players oftentimes assume that the numbers on the roulette wheel simply go in sequence from 0 to 36. In fact, the numbers are aligned in a seemingly random order around the wheel, and the sequence is different depending on whether you are playing European or American Roulette.

American Roulette Wheel
The American roulette wheel has all of the numbers from 1 to 36, plus a 0 and a 00 square, for a total of 38 possible numbers. The numbers are purposefully arranged in a non-sequential order. Originally this was done to avoid cheating on early roulette wheels (old fashioned roulette wheels often had a bias to one side of the wheel or the other, which would have made it easier to win if the player knew which set of numbers won more often).

Today, the placement of the numbers on the roulette wheel does not make a big difference in American roulette (roulette wheels are much more precise and a “biased” wheel is very rare in modern casinos). The order of the numbers going clockwise are as follows:

0, 28, 9, 26, 30, 11, 7, 20, 32, 17, 5, 22, 34, 15, 3, 24, 36, 13, 1, 00, 27, 10, 25, 29, 12, 8, 19, 31, 18, 6, 21, 33, 16, 4, 23, 35, 14, 2

European Roulette Wheel

Much like the American roulette wheel, the numbers on the European version are also ordered in a seemingly random sequence, but the order is completely different than the American Roulette version. The sequence of numbers on the European Roulette wheel are as follows:

0, 32, 15, 19, 4, 21, 2, 25, 17, 34, 6, 27, 13, 36, 11, 30, 8, 23, 10, 5, 24, 16, 33, 1, 20, 14, 31, 9, 22, 18, 29, 7, 28, 12, 35, 3, 26

However, unlike the American version, in European roulette the order of the numbers is important when it comes to betting. On the European roulette table, you will notice there is an extra section that is not on the American roulette table at all. It looks like this:

This section of the European roulette table offers a number of extra bets based on where the numbers are located on the European roulette wheel. The extra available bets are:

• Neighbor Bets: A neighbor bet is a wager placed on any number on the oval portion of the table shown above. This bet places a straight bet on the number selected, plus 2 numbers to the left and two numbers to the right as they are arranged on the European roulette wheel. For example, placing a neighbor bet on “0” would place a straight bet on 0, 3, 15, 26, and 32. If you look at the diagram of the European Roulette wheel above, you will see that the numbers 3, 15, 26, and 32 all surround the number 0 on the wheel.
• Voisins du Zero: This bet is a wager on 0, on the 9 numbers to the left of 0, and on the 7 numbers to the right of 0 as they arranged on the European roulette wheel. Bets are split between numbers that are next to each other on the rectangular table. 2 chips are placed on the 0, 2, 3 trio; 1 chip is placed on the 4-7 split; 1 chip is placed on the 12-15 split; 1 chip is placed on the 18-21 split; 1 chip is placed on the 19-22 split; 1 chip is placed on the 32-35 split; and 2 chips are placed on the corner of 25/26/28/29.
• Tiers du Cylinre: This bet is a wager on numbers that make up one third of the European roulette wheel (33, 16, 24, 5, 10, 23, 8, 30, 11, 36, 13, and 27). All of the bets are split between numbers that are next to each other on the rectangular section table.
• Orphans: This bet is a wager on the numbers 1, 6, 9, 14, 17, 20, 31, and 34. Bets are split between numbers that are next to one another on the rectangular table. The number 1 has a straight bet placed on it and the number 17 has two split bets placed on it; one with number 14 and another with number 20.

All of the above European roulette bets still pay out the same odds as any other bet in roulette. Making one of these bets is simply a way of placing a series of bets on a specific section of the wheel without having to do all of them individually.

If you want to learn more about how to play and win at roulette, check out the roulette strategy guide, or some of our other roulette articles: the difference between American and European Roulette, how to make roulette bets, all about roulette odds, the roulette glossary.

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Jeff has been an active gambler from the age of 18 when he won a small local poker tournament in San Jose, CA. He moved on to playing other casino games shortly thereafter, with a great interest in the statistics and mathematics involved. He has studied the historical gambling systems of nearly every popular casino game played today, and has written strategy books for several of them including blackjack, slot machines, craps, and roulette. Jeff has been a daily contributor to the Silver Oak Casino blog since the beginning of 2009.