What is the probability of getting a number divisible by 3 when a random number is chosen from 1 to 50?

Official Answer

Certainly! As a data scientist, I've frequently encountered the necessity to distill complex problems into manageable, quantifiable solutions, which directly applies to this probability question. In approaching this problem, I leverage both my analytical mindset and my proficiency in statistical analysis, ensuring my reasoning is not just theoretical but also practically grounded.

Let's dive into the problem. When we're looking at the numbers 1 to 50 and seeking those divisible by 3, we're essentially examining the series: 3, 6, 9, ..., 48. This series forms an arithmetic progression where the first term, (a_1), is 3, and the common difference, (d), is also 3. The task at hand is to find the total number of terms, (n), that are divisible by 3 within this range.

To find (n), we can use the formula for the (n)th term of an arithmetic progression: (a_n = a_1 + (n-1)d). Knowing the last term, (a_n), is 48, we can substitute and rearrange to solve for (n): [48 = 3 + (n-1)3] [45 = 3(n-1)] [15 = n-1] [n = 16]

Thus, there are 16 numbers between 1 and 50 that are divisible by 3.

Now, to find the probability of selecting one of these numbers at random, we apply the basic probability formula, where the probability is the number of favorable outcomes divided by the total number of outcomes. Here, the favorable outcomes are the 16 numbers divisible by 3, and the total number of outcomes is 50, the total numbers in our range.

[P(\text{divisible by 3}) = \frac{16}{50} = \frac{8}{25}]

Therefore, the probability of randomly selecting a number between 1 and 50 that is divisible by 3 is (\frac{8}{25}).

This kind of problem-solving is emblematic of the work I do as a data scientist. It's about breaking down the problem into its component parts, applying mathematical and statistical principles to find a solution, and clearly communicating the rationale behind the solution. In the realm of data science, whether it's analyzing data sets to glean insights or developing algorithms for predictive modeling, the core principles of logical structuring, problem-solving, and clear communication are paramount. This methodology not only ensures accuracy in my work but also facilitates effective collaboration with stakeholders to drive data-driven decision-making.