A jar contains 50 white and 50 black marbles. If you draw 2 marbles at random without replacement, what is the probability that they are the same color?

Official Answer

Certainly! Let's delve into this intriguing probability question by breaking it down step by step, ensuring that we fully understand each part of the process and how it connects to my background as a Data Scientist. The question at hand involves calculating the probability of drawing two marbles of the same color from a jar containing an equal number of white and black marbles, without replacement.

To begin with, the total number of ways to draw any two marbles from the jar is a classic combinatorial problem. With 100 marbles in total (50 white and 50 black), the number of ways to choose any two marbles is given by the combination formula (C(n, k) = \frac{n!}{k!(n-k)!}), where (n) is the total number of marbles, and (k) is the number of marbles we want to choose. Plugging in our values, we have (C(100, 2) = \frac{100!}{2!(98)!}), simplifying to 4,950 possible combinations.

Next, let's consider the probability of drawing two marbles of the same color. Since there are two possibilities—drawing two white marbles or drawing two black marbles—we'll calculate the probability for each scenario and then add them together. For drawing two white marbles, the number of favorable outcomes is (C(50, 2) = \frac{50!}{2!(48)!}), which simplifies to 1,225. Similarly, the probability of drawing two black marbles is the same, given the identical number of black marbles. Therefore, the total number of favorable outcomes for drawing two marbles of the same color is (2 \times 1,225 = 2,450).

To find the probability, we divide the number of favorable outcomes by the total number of outcomes. Thus, the probability of drawing two marbles of the same color is (\frac{2,450}{4,950}), which simplifies to (\frac{1}{2}) or 50%.

In my role as a Data Scientist, dissecting problems like this is part of my daily routine. Whether it's analyzing data sets to predict consumer behavior, developing algorithms for machine learning models, or optimizing data visualization for stakeholder presentations, the foundational principles of probability underpin much of my work. This question, while seemingly straightforward, encapsulates the essence of problem-solving: breaking down complex problems into manageable parts, applying mathematical principles, and drawing actionable insights.

Throughout my career, I've harnessed the power of probability and statistics not only to solve theoretical problems but to drive real-world decisions. For instance, by analyzing user engagement data, I've been able to predict churn rates and inform targeted retention strategies. Similarly, in machine learning projects, understanding the underlying probabilities helps in fine-tuning models for better accuracy. This problem-solving approach, rooted in a deep understanding of probability, has been a cornerstone of my success as a Data Scientist.

I believe this precise and methodical approach to problem-solving would be a valuable asset to your team. Not only do I bring a strong foundation in probability and statistics, but I also have a proven track record of applying these principles to drive impactful business outcomes. I'm eager to leverage my skills and experiences to contribute to your team's success.