If a jar contains 3 red, 4 green, and 5 blue marbles, what is the probability of drawing a green marble, then a blue marble, with replacement?

Official Answer

Certainly, approaching a probability question like this requires a clear understanding of the concepts of probability, especially when it comes to events with replacement. As a Data Scientist, my approach to problem-solving involves breaking down the problem into manageable parts, applying statistical theories, and then synthesizing the information to arrive at a solution. Let's delve into how I would solve this specific probability question.

First, let's understand the scenario. We have a jar containing 3 red, 4 green, and 5 blue marbles, making a total of 12 marbles. The question asks for the probability of drawing a green marble followed by a blue marble, with the condition that after drawing a marble, it is replaced back into the jar before the next draw. This replacement ensures that the total number of marbles in the jar remains constant across both draws.

The probability of drawing a green marble from the jar, given that there are 4 green marbles out of a total of 12, is calculated as the number of favorable outcomes (drawing a green marble) divided by the total number of outcomes (total marbles). Thus, the probability of the first event (drawing a green marble) is 4/12, which simplifies to 1/3.

Moving on to the second part of the question, since the drawn marble is replaced, the total number of marbles in the jar remains at 12. Now, we're interested in drawing a blue marble. There are 5 blue marbles out of the total 12 marbles. The probability of this event (drawing a blue marble) is therefore 5/12.

To find the overall probability of both events happening in sequence (drawing a green marble and then a blue marble with replacement), we multiply the probabilities of the individual events. This is because, in probability theory, the probability of two independent events both occurring is the product of their individual probabilities.

Hence, the probability of drawing a green marble followed by a blue marble, with replacement, is (1/3) * (5/12). Multiplying these fractions gives us 5/36.

In my work as a Data Scientist, dealing with probabilities is part of daily tasks, whether it's evaluating the likelihood of certain outcomes based on historical data, or creating models that predict future trends. The key is always to break down the problem, apply the relevant statistical principles, and communicate the solution in a clear, understandable manner. This approach not only ensures accuracy but also enhances the decision-making process by providing a quantitative basis for predictions and analyses.

This example demonstrates how, by understanding and applying fundamental probability principles, we can solve complex problems systematically. For job seekers in data science and related fields, showcasing this type of structured problem-solving ability is crucial, not only in interviews but also in practical, real-world applications.