Rules of Roulette:
Choose a number, any number. Heck, choose as many numbers as you want! This is the main premise of a traditional roulette table if it were to be told to a 5-year-old. 

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Considering that you bet x amount of money per number (      =-x), your prize increases as you choose less numbers in this order:

• 1 number: 36x per number (/n)

• 2 numbers: 18x/n

• 3 numbers: 12x/n

• 4 numbers: 9x/n

• 6 numbers: 6x/n

• 12 numbers: 3x/n

• 18 numbers: 2x/n

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Expectancy:
Now let’s delve into a lesson in probability.        is the probability of an event x happening, let it be the chance of raining today or the chance of scoring in a match. In this case, it will be the chance of the ball falling on one of the betted numbers. Expectancy or      is however equal to      multiplied by the payout of the event (      ). If the expectancy of an event is positive, you should participate in it. If it is negative, you shouldn’t. And if it is 0, it doesn’t matter if you participate in it or not. 

 

For every strategy, we can calculate their expectancies by this formula:

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where 1-      is the possibility of an event not happening and               being the payout of this event not happening. As an example, let’s calculate the expectancy for betting on 8:

 

 

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and

 

 

 

 

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Thus: 

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Or, if you were to choose the 2nd twelve numbers [13,24]:

 

 

 

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and

 

 

 

 

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Thus:

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Since the possibility of the ball falling to any number including 0 is 1/37, all other strategies will have the same expectancy, -x/37 where x is the amount of money you bet on each number you choose. Due to this, if you play roulette for a really long time, say for 1000 rounds and you bet 10 TL for each bet (x=10), your expectancy will be -10/37TL, or you will lose 10/37TL on average for every game, resulting in a loss of 10000/37TL or 270.3 TL in total (     =-270.73 TL). Thus, you shouldn’t play the roulette. 

 

Other Uses:
Expectancies are one of the many properties that are so common, yet go totally unnoticed in daily life. Every decision we make depends on expectancies. As an example, while crossing a road, you do so if it is safe enough to cross. The safety depends on the chance of you crossing the road in time, which is the possibility, and the damage dealt to you and the environment if you can cross in time (no damage) or not (car crash that can kill you), which is the payout. They are also popular in other types of gambling or games like Poker, Blackjack, and more.

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Sources:

32 inç rulet tekerleği katı ahşap Casino rulet profesyonel. 2023. Aliexpress, https://shorturl.at/fjAEK. Accessed 19 Nov. 2023. 
Kanon. “Rulet Nedir, Nasıl Oynanır, Kazançlı Mıdır? - Ruletin Matematiği.” Donanım Arşivi, 7 Apr. 2023, https://forum.donanimarsivi.com/konu/rulet-nedir-nasil-oynanir-kazancli-midir-ruletin-matematigi.718705/#post-6822351. Accessed 19 Nov. 2023. 
Nesin Vakfı Matematik Köyü, 2021, 6th Term, Lectures that the author joined.