A bag contains 4 white, 5 black, and 6 red balls. Two balls are drawn randomly. What is the probability that they are of different colors?

Official Answer

Certainly, approaching a probability question like this, especially in the context of a Data Scientist interview, allows me to showcase not just my analytical capabilities but also how I break down and communicate complex problems in a comprehensible manner. Let's delve into solving this problem, step by step.

Initially, we acknowledge the total number of balls in the bag, which sums up to 15 (4 white, 5 black, and 6 red). The question at hand is to determine the probability of drawing two balls of different colors.

To tackle this, I consider the various combinations through which we can achieve this outcome:

1. Drawing a White then a Black ball: The probability of drawing a white ball first is 4 out of 15. Assuming a white ball has been drawn, we now have 14 balls left, with 5 of them being black. Hence, the probability of then drawing a black ball is 5 out of 14.

   > The combined probability for this scenario is (4/15) * (5/14).
   

2. Drawing a White then a Red ball: Following a similar logic, if a white ball is drawn first, the probability of drawing a red ball next is 6 out of 14 (since one white ball is already out).

   > Thus, the probability for this combination is (4/15) * (6/14).
   

3. Drawing a Black then a White ball: If we start by drawing a black ball, the probability is 5 out of 15. To then draw a white ball, with one black ball out, is 4 out of 14.

   > This gives us a probability of (5/15) * (4/14).
   

4. Drawing a Black then a Red ball: Starting with a black ball, the subsequent probability of drawing a red ball is 6 out of 14.

   > So, the probability for this case is (5/15) * (6/14).
   

5. Drawing a Red then a White ball: Starting with a red ball, the probability of drawing a white ball next is 4 out of 14.

   > The combined probability here is (6/15) * (4/14).
   

6. Drawing a Red then a Black ball: Lastly, if a red ball is drawn first, drawing a black ball next has a probability of 5 out of 14.

   > Hence, this probability is (6/15) * (5/14).
   

Now, to find the total probability of drawing two balls of different colors, we sum up the probabilities of all these scenarios:

Total Probability = (4/15 * 5/14) + (4/15 * 6/14) + (5/15 * 4/14) + (5/15 * 6/14) + (6/15 * 4/14) + (6/15 * 5/14)

Calculating the above gives us the total probability. This methodical approach not only ensures accuracy but also demonstrates a logical problem-solving strategy, crucial for a Data Scientist. It's about breaking down the problem into smaller, manageable parts and then synthesizing the information to arrive at a solution.

In my previous projects, such as when analyzing user behavior data for insights into product improvement or when optimizing algorithms for better performance, this approach has always been at the core of my methodology. It's about being meticulous, analytical, and, most importantly, clear in communication.

In essence, this probability question is not just about finding the right answer. It's about demonstrating a mindset—a way of thinking and problem-solving that's critical in data science. It’s about showing how we can take complex, real-world problems, deconstruct them into their constituent elements, analyze those elements, and synthesize our findings into actionable insights.