# SOCR EduMaterials Activities RouletteExperiment

### From Socr

## Contents |

## Roulette Experiment

## Description

The American Roulette wheel has 38 slots numbered 00, 0, and 1-36. Slots 00 and 0 are green. Half of the slots numbered 1-36 are red and half are black. The experiment consists of rolling a ball in a groove in the wheel; the ball eventually falls randomly into one of the 38 slots. The roulette wheel is shown in the left graph panel; the ball is shown on each update.

One of seven different bets can be selected from the list box:

Bet on 1: this is an example of a straight bet, and the bet pays 35:1

Bet on 1, 2: this is an example of a split bet, and pays 17:1

Bet on 1, 2, 3: this is an example of a row bet, and bet pays 11:1

Bet on 1, 2, 4, 5: this is an example of 4-number bet, and pays 8:1

Bet on 1-6: this is an example of a 2-row bet, and pays 5:1

Bet on 1-12: this is an example of a 12-number bet, and pays 2:1

Bet on 1-18: this is an example of an 18-number bet, and pays 1:1

On each update, the outcome X is shown graphically in the first panel and recorded numerically in the first table. Random variable W gives the net winnings for the chosen bet; this variable is recorded in the first table on each update. The density function and moments of W are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density and moments of W are shown in red in the distribution graph and are recorded in the distribution table.

## Goal

The Roulette Experiments provides a real-life simulation of the American Roulette game. It demonstrates the probabilities and outcomes for every trial under different conditions. By using this applet, users should be able to have a better understanding of the game, Roulette.

## Experiment

Go to SOCR Experiments and select the **Roulette Experiment** from the drop-down list of experiments on the top left. The image below shows the initial view of this experiment:

When pressing the play button, one trial will be executed and recorded in the distribution table below. The fast forward button symbolizes the nth number of trials to be executed each time. The stop button ceases any activity and is helpful when the experimenter chooses “continuous,” indicating an infinite number of events. The fourth button will reset the entire experiment, deleting all previous information and data collected. The “update” scroll indicates nth number of trials (1, 10, 100, or 1000) performed when selecting the fast forward button and the “stop” scroll indicates the maximum number of trials in the experiment.

Bet on 1

The distribution graph illustrates a high probability for values of 1 and a low value for 35. As the number of trials increase, the empirical density graph will converge to the distribution graph as demonstrated below:

Bet on 1, 2

Similarly, the distribution graph is most saturated on the left side with a very small proportion for the highest value in the x axis. As the number of trials increase, the empirical density graph will converge to the distribution graph as demonstrated below:

While going down the list box and selecting the specific bet, the first value on the x axis obtains a much higher proportion than the last value on the x axis for every distribution graph. But as the ratio of pay increases, the distribution graph begins to shift until Bet 1-18, in which the first and last values are equal. This is shown in the image below:

Since the distribution graph represents the chances of winning, users should understand that the lower the proportion value is on the right side of the graph, the lower their chances are of winning. That is why selecting Bet 1-18 has an equal chance of winning and losing with a graph illustrating equal values for -1, 1.

## Applications

The Roulette Experiment is best applicable in gambling situations such as:

A gambler would like to illustrate the chances of winning in the American Roulette game.

A player wants to determine which of the Bets will give him the highest chance of winning after ten trials.

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