If you roll four six-sided dice, what is the probability that at least two dice will land on the same number?

Official Answer

Certainly, approaching a probability question like this, especially in a high-stakes interview, offers a fantastic opportunity to showcase not just my technical expertise but also my problem-solving approach which I have honed over years of experience as a Data Scientist. So, let me walk you through my thought process.

First, let's consider the complement of the event in question, as it's often easier to calculate. The opposite of having at least two dice showing the same number would be all dice showing different numbers. Since we are dealing with six-sided dice and we have four of them, let's calculate the probability of all dice showing a unique number.

For the first die, any face value would work, so it has 6 possible outcomes out of 6, which gives us a probability of ( \frac{6}{6} = 1 ). For the second die to not match the first die, it can land on any of the 5 remaining numbers, giving us a probability of ( \frac{5}{6} ). For the third die, we need it to be different from the first two, so it has 4 out of 6 possible outcomes, resulting in ( \frac{4}{6} ). Finally, for the fourth die, it must be different from the first three, leaving 3 out of 6 possible outcomes or ( \frac{3}{6} ).

Multiplying these probabilities gives us the overall probability of all dice showing different numbers: ( 1 \times \frac{5}{6} \times \frac{4}{6} \times \frac{3}{6} = \frac{5}{54} ).

However, we wanted the probability of at least two dice showing the same number. Since we've calculated the opposite, we'll subtract our result from 1 to get the desired probability: ( 1 - \frac{5}{54} ).

Simplifying this gives us ( \frac{49}{54} ). So, the probability that at least two of the four six-sided dice will land on the same number is ( \frac{49}{54} ).

In presenting this solution, what I aimed to showcase was not just the mathematical computation but the logical strategy behind tackling probability questions, which is crucial in data science. This approach of breaking down problems, considering complements when they offer an easier path, and systematically applying probability principles reflects the analytical mindset I bring to my role. It exemplifies how I navigate data analysis, model building, and even complex problem-solving in real-world scenarios. This perspective allows me to bring clarity to data-driven insights, ensuring they are not only accurate but also actionable and understandable to stakeholders across the board.