# SOCR EduMaterials Activities RedAndBlackExperiment

## Red and Black Experiment

## Description

In the red and black experiment, a player starts with an initial fortune *x* and bets (at even stakes) on independent trials for which the probability of winning is *p*. Play continues until the player is either ruined or reaches a fixed target fortune a. Either of two player strategies can be selected from a list box. With timid play, the player bets 1 on each trial. With bold play, the player bets her entire fortune or what is needed to reach the target (whichever is smaller).

Variable J indicates the events that the player wins (reaches her target) and variable N is the number of trials played. These variables are recorded on each update. The first graph shows the initial and target fortunes and the final outcome. The density of J is shown in the middle graph and table, and the mean of N is shown in the last graph and table. The parameters *x* and *p* can be varied with scroll bars, and the target fortune *a* can be chosen from a list box.

## Goal

The Red and Black Experiment is a basic gambling model in which it demonstrates the probabilities of outcomes if the trials are unfair. By using this applet, users will be able to grasp a better understanding of how some casino games work like roulette, or a generalization involving biased probabilities.

## Experiment

Go to the SOCR Experiment and select the Red and Black 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.

When varying parameters *x* and *p*, increasing either one or both of them will increase the red values in the first graph and it will also increase the proportion of value 1 in the middle graph. When decreasing the parameters, the opposite will occur in which the red portion will decrease as well as the proportion for value 1, thus increasing the proportion value of 0 in the middle graph. The image shown below demonstrates this as the value of *x* and *p* obtains high values:

As for parameter *a*, when it has a large value, the red portion in the first graph is small whereas having a small value, the red portion is large. When the red portion in the first graph is large, the distribution graph in the third graph will be very small. So, when parameter *a* has a large value, the distribution graph in the third image will be very large. The image below demonstrates this:

## Applications

The Red and Black Experiment is best applicable in gambling situations such as:

A gambler would like to maximize the probability of reaching the target of fortune. By using this applet, he is able to test on Timid or Bold play. Afterwards he will be able to draw personal conclusions as to which approach will better his chances of winning.

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