What is the probability that a five-digit number formed by randomly arranging the digits 1, 2, 3, 4, and 5 is divisible by 4?

Official Answer

Certainly! To tackle this probability question effectively, let me bring into play my background as a Data Scientist, intertwining my analytical prowess with a structured approach to problem-solving.

For a five-digit number to be divisible by 4, the number formed by its last two digits must be divisible by 4. In our scenario, we're working with the digits 1, 2, 3, 4, and 5. Let's examine the pairs that can form a number divisible by 4: 12, 24, 32, and 52. Notice how we focus on the combinations of the last two digits because, according to the divisibility rules for 4, the divisibility of the entire number hinges on its last two digits.

Given this insight, we have 4 valid pairs (12, 24, 32, 52) that meet our criteria. Now, for each of these pairs, the remaining three digits can be arranged in 3! ways (since once we've selected our last two digits, there are three digits left, which can be arranged in any order). So, for each pair, we have 3! or 6 arrangements.

Therefore, with 4 valid pairs and 6 arrangements for each, we have a total of 4 * 6 = 24 valid five-digit numbers that are divisible by 4.

Now, to find the probability, we need to consider the total number of unique five-digit numbers that can be formed with the digits 1, 2, 3, 4, and 5. Since each digit is unique and must be used exactly once, we have a total of 5! arrangements. That is, 5 * 4 * 3 * 2 * 1 = 120 possible five-digit numbers.

The probability we seek is the ratio of the number of valid five-digit numbers divisible by 4 to the total number of five-digit numbers. That is, the probability = 24 / 120.

This simplifies to 1 / 5.

Thus, the probability that a randomly arranged five-digit number formed from the digits 1, 2, 3, 4, and 5 is divisible by 4 is 1/5 or 20%.

In presenting this solution, I leveraged a methodical approach, breaking down the problem into manageable parts—identifying valid digit pairs and calculating permutations. This mirrors the analytical strategies I apply in data science: dissect complex data problems, utilize mathematical principles, and articulate clear, actionable insights. This approach not only ensures accuracy but also enhances the interpretability of data-driven solutions, empowering stakeholders to make informed decisions.