Probability of Drawing Two Blue Marbles With Replacement

What is the probability of selecting two blue marbles with replacement?

This problem focuses on calculating probabilities in a discrete probability space, which is a foundational concept in the study of probability. When dealing with probabilities, especially in discrete mathematics, it's important to understand the difference between events that occur with replacement and those that occur without replacement. In this case, the phrase 'with replacement' indicates that the outcome of the first event does not affect the second event, as the marble is replaced back into the mix. Thus, the events are considered independent.

When analyzing problems like this, it's crucial to apply the principles of multiplication rule of probability for independent events. This rule states that the probability of two independent events both occurring is the product of their individual probabilities. Hence, understanding how to identify independence between events and correctly applying this rule is essential when solving problems in discrete probability.

Additionally, this problem serves as an introduction to more complex problems where you deal with multiple events and calculate combined probabilities. It lays the groundwork for exploring conditional probabilities and other deeper aspects like expectation and variance in future scenarios. These concepts are important as they apply not just to theoretical problems, but also to practical real-world situations, especially in fields like computer science, data science, and operations research.