A game involves rolling a fair four-sided die and a fair six-sided die together. What is the probability that the sum of the two dice is odd?

Official Answer

Certainly, approaching a question that involves probability, especially in the context of dice, requires a clear understanding of the possible outcomes and how they align with the condition specified—in this case, achieving an odd sum from the roll of a four-sided die and a six-sided die.

First, let's recognize that a four-sided die has four possible outcomes [1, 2, 3, 4], and a six-sided die has six possible outcomes [1, 2, 3, 4, 5, 6]. The sum of the two dice can range from 2 (1+1) to 10 (4+6). Now, for the sum to be odd, one die must land on an even number and the other on an odd number, as even + odd = odd.

Let's break down the outcomes. For the four-sided die, there are two even outcomes (2, 4) and two odd outcomes (1, 3). Similarly, the six-sided die has three even outcomes (2, 4, 6) and three odd outcomes (1, 3, 5).

Now, to calculate the probability, we need to consider the combinations that lead to an odd sum. We can pair each even outcome from one die with each odd outcome from the other die.

• For the four-sided die landing on an even number (2 possibilities) and the six-sided die landing on an odd number (3 possibilities), we have 2 * 3 = 6 favorable outcomes.
• Conversely, for the four-sided die landing on an odd number (2 possibilities) and the six-sided die landing on an even number (3 possibilities), we again have 2 * 3 = 6 favorable outcomes.

Adding these together, we have a total of 6 + 6 = 12 favorable outcomes for achieving an odd sum.

To find the probability, we need to divide the number of favorable outcomes by the total number of possible outcomes. The total number of possible outcomes when rolling the two dice is the product of their individual outcomes, which is 4 * 6 = 24.

Therefore, the probability of rolling an odd sum is 12 / 24 = 1/2 or 50%.

As a Data Scientist, approaching this problem efficiently involved breaking down the problem into smaller, manageable parts and applying basic principles of combinatorics. This approach not only simplifies complex problems but also demonstrates a methodical mindset, crucial in data analysis and model building. It's the kind of structured yet flexible thinking that I've applied in various projects, notably in developing predictive models where understanding and calculating probabilities are fundamental. This methodology allows for a clear communication of complex analytical findings to stakeholders, an essential skill in bridging the gap between data science and business strategy.