Mathematics and Roulette

Roulette is a French word for small wheel. It was invented by great French mathematician, Pascal. It involves spinning a wheel and dropping a small ball along the edge. A roulette contains 37 or 38 separately numbered pockets in which a ball must land. The main pockets are numbered from 1 to 36 alternate between red and black but not arranged in numerical order.

When the wheel stops, the ball will have found its way into one of the slots. The 38 slots contain the numbers 1 through 36 and the special numbers 0 and 00(except in European casinos, its roulette does not have number 00) 18 number are in red while another 18 are in black. The special numbers 0 and OO are in green. If a player bets on a single number and wins, the payout is 35:1, meaning that for each dollar bet, one wins an additional $35.00.

Board depiction (American Roulette)

We know that the probability of an event E is defines as :

P (E) = Number of way E can happen / Total numbers of probabilities

For example, the probability of ball landing in black is :

P (B) = 18/38

= 9/19

Below is a list of the probabilities of winning for each of the other bets in Roulette.
1 Number "Straight Up": 1/38
2 Numbers "Split" or "0/00": 2/38 = 1/19
3 Numbers "Street": 3/38
4 Numbers "Corner": 4/38 = 2/19
5 Numbers "First Five": 5/38
6 Numbers "Line": 6/38 = 3/19
12 Numbers "Dozen" and Columns: 12/38 = 6/19
18 Numbers "Red/Black," "High/Low", "Odd/Even": 18/38 = 9/19

• Straight: a single number. The chip is placed entirely on this square and no other.
• Split: a bet on two adjoining numbers, either on the vertical or horizontal (as in 14-17 or 8-9). The chip is placed on the edge between these numbers.
• Street: a bet on three numbers on a single horizontal line. The chip is placed on the edge of the number at the left or the right, depending on the layout
• Corner (or square): a bet on four numbers in a square layout (as in 11-12-14-15). The chip is placed at the horizontal and vertical intersection of these numbers.
• Sixline (or 'sixaine'): a bet on two adjoining streets, with the chip placed at the corresponding intersection (as in 25-30).
• Column: a bet on all 12 numbers on any of the three vertical lines (such as 1-4-7-10 on down to 34). The chip is placed on the space below the final number in this string.

It is essential to understand how good or bad a bet is. We talk about how good the bet is by saying that if you make the same bet many many times, how much will you expect to win (or lose) per bet on average. We will exhibit how to calculate the expected value with an example.

For Example :
Find the expected value for "Split" bet that pays 17 to 1 if you wager $100.

Solution
First note that there are two possibilities. The first is that one of your two numbers is hit. In this case you win $1700. The second is that a different number is hit. In this case you win -100 dollars. To find the expected value, multiply the financial result by the probability of that happening. Then add all of these products.

Since P(Win) = 1/19 and a win results in 1700, we multiply
(1/19)(1700) = 1700/19

To find the probability of losing, just subtract the probability of winning from 1 to get
P(Lose) = 1 - P(Win) = 1 - 1/19 = 18/19.

Losing results in winning -100, so we multiply
(18/19)(-100) = -1800/19

finally add these partial results to get
1700/19 + (-1800/19) = -100/19 = -5.26

Now to interpret this number. If you make the "Split" bet many times at $100 per wager, you will average a loss of $5.26 per wager.

Note : Bet at your on risk. This blogger bear no responsibility on your wins and loses.