In a box of 12 pens, 4 are blue, 5 are black, and 3 are red. If you select 3 pens at random, what is the probability that you get pens of all different colors?

Official Answer

Certainly, approaching a probability question like this, it's essential to systematically break down the problem and apply foundational principles of combinatorics and probability theory, areas I've deeply engaged with in my role as a Data Scientist. In dealing with large datasets and constructing predictive models, understanding and applying probability has been crucial. So, let me walk you through how I'd tackle this problem, leveraging my experience in data analysis and model building to provide a structured solution.

First, let's consider the total number of ways we can select 3 pens from the 12 available, without any regard to their color. This is a classic combination problem where order does not matter, and we can use the combination formula nCr = n! / [r!(n-r)!], where n is the total number of items, and r is the number of items to choose. For our scenario, n=12 pens, and r=3 pens we are choosing, giving us a total of 12C3 = 12! / [3!(12-3)!] = 220 ways to choose any 3 pens.

Next, to address the specific requirement of selecting 3 pens of all different colors, we need to recognize that we are dealing with a multi-step selection process. We select one pen of each color, which involves choosing 1 out of the 4 blue pens, 1 out of the 5 black pens, and 1 out of the 3 red pens. Each of these selections is independent, and thus, we can calculate the number of ways to select one pen of each color by multiplying the number of ways to choose one pen from each color group together: 4C1 * 5C1 * 3C1. Calculating this gives us 4 * 5 * 3 = 60 ways to select one pen of each color.

To find the probability of selecting 3 pens of all different colors from the box, we then take the number of ways to successfully meet our condition (60 ways) and divide it by the total number of ways to choose any 3 pens (220 ways). This gives us a probability of 60 / 220, which simplifies to 3 / 11.

Throughout my career, whether it's been analyzing user behavior, optimizing algorithms, or conducting experimental research, the ability to deconstruct problems into their fundamental parts and apply logical, mathematical reasoning has been indispensable. This question is a great example of how foundational mathematical concepts are applied in real-world data science scenarios. It’s not just about calculating probabilities in isolation; it’s about understanding the underlying distributions, relationships, and dynamics. In modeling work, this approach helps in making informed decisions based on probabilistic outcomes, which is essential for predictive accuracy and robustness.

In presenting this solution during an interview, the key is not just in arriving at the correct answer, but in clearly articulating the thought process that led there. This demonstrates not only technical proficiency but also analytical thinking and problem-solving skills—qualities that are invaluable in any data science role.