Some notes taken from Prof. Guttag excellent discussion on the topic.

The technique was first developed by Stanislaw Ulam, a mathematician who worked on the Manhattan Project.

A method of estimating the value of an unknown quantity using the principles of inferential statistics.

Inferential Statistics

• Population : Set of examples
• Sample : proper subset of population
• Random sample tends to exhibit the same qualities as the population.

Confidence depends on:

• sample size
• variance. As variance grows, larger samples are required to have the same degree of confidence.

Roulette Considerations

• Law of large numbers (Bernoulli’s law) If the probability is p, the difference between the prob obtained by samples to p goes to 0 as number of samples goes to infinity.

• Gambler’s Fallacy and Regression to the mean

Gambler’s Fallacy: if a particular event occurs more frequently than normal during the past it is less likely to happen in the future (or vice versa), when it has otherwise been established that the probability of such events does not depend on what has happened in the past.

Regression to the mean: Following an extreme random event, it is likely that the next random event will be less extreme.

Quantifying Variation

$$ variance(X) = \frac{\sum_{x\in X}(x - \mu)^2}{|X|} $$

The mean $$\mu$$

$$\sigma(x) = \sqrt{variance(X)}$$

• outliers have a big effect
• standard deviation always considered relative to mean

Empirical Rule

• approx. 68% of data within one standard deviation of mean
• approx. 95% of data within 1.96 standard deviation of mean - usually used
• approx. 99.7% of data within 3 standard deviation of mean

Assumptions

• mean estimation error is zero. Therefore no bias
• The distribution of errors in estimates is normal (mean=zero, sd=1)

Probability Density Function

• Distributions defined by Probability Density Function, PDF
• Probability of a random value lying between two values
• Defines a curve where the range in the X-axis is between the maximum and minimum values of the variable.
• Area under curve between two points defined the probability of an example falling in that range

import random

class FairRoulette():
    def __init__(self):
        self.pockets = []
        for i in range(1, 37):
            self.pockets.append(i)
        
        self.ball=None

        self.pocket_odds = len(self.pockets) - 1

        random.seed(0)

    def spin(self):
        self.ball = random.choice(self.pockets)

    def bet_pocket(self, pocket, amount):
        '''
        pocket: pocket placing bet
        amount: sum being bet
        '''
        if str(pocket) == str(self.ball):
            return amount * self.pocket_odds
        else:
            return -amount
    def __str__(self) -> str:
        return 'fair roulette'

class EURoulette(FairRoulette):
    def __init__(self):
        super().__init__()
        self.pockets.append('O')
    def __str__(self) -> str:
        return 'EU Roulette'

class USRoulette(EURoulette):
    def __init__(self):
        super().__init__()
        self.pockets.append('OO')
    def __str__(self) -> str:
        return 'US Roulette'


def play_roulette(game, num_spins, pocket, bet):
    '''
    Arguments:
    game: Roulette game being played
    num_spins: number of spins for the simulation
    pocket: pocket placing bet
    bet: amount of bet
    '''
    total_pocket = 0
    for i in range(num_spins):
        game.spin()
        total_pocket += game.bet_pocket(pocket, bet)

    print(f'{num_spins} spins of {game}')
    print(f'expected return betting {pocket} = {str(100*total_pocket/num_spins)}%')

    return total_pocket/num_spins

if __name__ == "__main__":
    game = FairRoulette()
    for num_spins in (100, 1000000):
        for i in range(3):
            # betting 1 dollar on number 2 for num-spins trials
            play_roulette(game, num_spins, 2, 1)