Marble Madness: Exploring the Sample Space and Probability of Marble Selection

 A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from the box. What is the sample space? If, at all times, each marble in the box is equally likely to be selected, what is the probability of each point in the sample space?

In this blog post, we'll be diving into the world of probability and statistics through a simple yet fun experiment. Imagine a box containing three marbles: one red, one green, and one blue. We'll be conducting an experiment that consists of taking one marble from the box, replacing it, and then drawing a second marble from the box.

First, let's explore the sample space of this experiment. The sample space, also known as the set of all possible outcomes, can be represented as a list of all possible combinations of two marbles. In this case, the sample space is:

{(red, red), (red, green), (red, blue), (green, red), (green, green), (green, blue), (blue, red), (blue, green), (blue, blue)}

Now, let's calculate the probability of each point in the sample space. Since each marble in the box is equally likely to be selected at all times, the probability of selecting a red marble is 1/3, a green marble is 1/3, and a blue marble is 1/3. Therefore, the probability of selecting a red marble for the first draw and a green marble for the second draw is (1/3) x (1/3) = 1/9. Similarly, the probability of selecting a blue marble for the first draw and a red marble for the second draw is also 1/9.

By using this method, we can calculate the probability of each point in the sample space. It's important to note that since the marbles are being replaced after each draw, the order in which they are selected doesn't matter, so the sample space contains only 9 combinations rather than 18. This is an example of sampling without replacement.

In conclusion, through this simple experiment, we've explored the concept of sample space and probability in a fun and interactive way. It's a great way to understand the fundamental concepts of probability and statistics and how they can be applied to real-world situations.