Roulette – An Analysis

Roulette – An Analysis

Roulette is played by spinning a ball around a wheel.  The wheel contains 37 to 38 numbered pockets from 0 to 36 for a 37-numbered wheel and 00 and 0 to 36 for a 38-numbered wheel.

The winning number is decided by which numbered pocket the ball comes to rest on.  Payment is then made based on where a player has wagered on a layout displaying all the numbers on the wheel.

This table assesses the wagers for Single-Zero (European) Roulette:

European Roulette (Single Zero)
Wager	Numbers covered	Probability (Win)	Probability (Loss)	Expectation	Break-even odds
Straight Up	1	0.02703	0.97297	-0.94595	36.00
Split	2	0.05405	0.94595	-0.89189	17.50
Street	3	0.08108	0.91892	-0.83784	11.33
Corner	4	0.10811	0.89189	-0.78378	8.25
6-Line	6	0.16216	0.83784	-0.67568	5.17
Dozen	12	0.32432	0.67568	-0.35135	2.08
Column	12	0.32432	0.67568	-0.35135	2.08
Small (1-18)	18	0.48649	0.51351	-0.02703	1.06
Big (19-36)	18	0.48649	0.51351	-0.02703	1.06
Red	18	0.48649	0.51351	-0.02703	1.06
Black	18	0.48649	0.51351	-0.02703	1.06
Even	18	0.48649	0.51351	-0.02703	1.06
Odd	18	0.48649	0.51351	-0.02703	1.06

This table assesses the wagers for Double-Zero (American) Roulette:

American Roulette (Double Zero)
Wager	Numbers covered	Probability (Win)	Probability (Loss)	Expectation	Break-even odds
Straight Up	1	0.02632	0.97368	-0.94737	37.00
Split	2	0.05263	0.94737	-0.89474	18.00
Street	3	0.07895	0.92105	-0.84211	11.67
Corner	4	0.10526	0.89474	-0.78947	8.50
6-Line	6	0.15789	0.84211	-0.68421	5.33
Dozen	12	0.31579	0.68421	-0.36842	2.17
Column	12	0.31579	0.68421	-0.36842	2.17
Small (1-18)	18	0.47368	0.52632	-0.05263	1.11
Big (19-36)	18	0.47368	0.52632	-0.05263	1.11
Red	18	0.47368	0.52632	-0.05263	1.11
Black	18	0.47368	0.52632	-0.05263	1.11
Even	18	0.47368	0.52632	-0.05263	1.11
Odd	18	0.47368	0.52632	-0.05263	1.11

Wager lists the specific wager type players can make on the layout.

Numbers covered indicates how many numbers the wager will be counted to be wagered on.  If the ball rests on any of the numbers covered by this wager, the wager is paid.

Probability (Win) indicates the probability that this wager will win. This is calculated by:

= Numbers covered/total numbers on the layout (37 for Single-Zero and 38 for Double-Zero)

Example:

For a Corner (Single-Zero) wager, 4/37 = 0.10811

Probability (Loss) indicates the probability that the wager will lose.  The calculation for this is 1-Probability (Win).

Example:

For a Street (Double-Zero) wager, 3/38 = 1-0.07895 = 0.92105

Expectation is the Probability (Win) – Probability (Loss).

Example:

For a Street (Double-Zero) wager, 0.07895 – 0.92105= -0.84211

Break-even odds indicates the odds at which the expectation for the house is 0.  The house should ALWAYS pay BELOW these odds to maintain a positive expectation.