A number is chosen at random from the first 100 positive integers, that is, from 1 to 100 inclusive. What is the probability that the selected number is divisible by 4?

Introduction / Context:
This question illustrates probability over a finite uniform sample space of integers. We choose one number at random from the first 100 positive integers and want the probability that it is divisible by 4. This involves simple counting of multiples of 4 within a specified range and then forming a probability ratio.

Given Data / Assumptions:

• The sample space consists of integers from 1 to 100 inclusive.
• All 100 integers are equally likely to be selected.
• We are interested in numbers that are divisible by 4, that is, numbers of the form 4k with integer k.
• The probability is favorable outcomes divided by total outcomes.

Concept / Approach:
The key idea is to count how many integers between 1 and 100 are multiples of 4. These form an arithmetic progression starting from 4 up to the largest multiple less than or equal to 100. Once we know this count, we divide by 100 to obtain the probability. This is a standard method for problems involving divisibility in a finite range.

Step-by-Step Solution:
Step 1: The total number of possible outcomes is 100, since we are selecting from 1 to 100 inclusive.Step 2: Identify the smallest positive integer divisible by 4 in this range, which is 4.Step 3: Identify the largest integer in the range 1 to 100 that is divisible by 4. The largest such number is 100, because 4 * 25 = 100.Step 4: The multiples of 4 in this range are 4, 8, 12, and so on up to 100. They form an arithmetic progression with first term 4 and common difference 4.Step 5: The number of terms in this progression is 25, since 4 * 25 = 100.Step 6: Therefore, there are 25 favorable outcomes, and the probability that the chosen number is divisible by 4 is 25 / 100 = 1 / 4.
Verification / Alternative check:
We can verify by observing that every group of four consecutive integers, such as 1 to 4 or 5 to 8, contains exactly one multiple of 4. The interval from 1 to 100 consists of 25 such groups. Hence there are exactly 25 numbers divisible by 4, which confirms the earlier count. This also aligns with the simple formula range size divided by step size to count equally spaced multiples.

Why Other Options Are Wrong:
2 and 1 are impossible as probabilities because they exceed the maximum possible value of 1. 1/2 would require 50 favorable outcomes, meaning every second number is a multiple of 4, which is not true. 3/10 would correspond to 30 favorable outcomes, but we have seen that there are exactly 25 multiples of 4 in the range, not 30.

Common Pitfalls:
Students sometimes miscount the multiples by including 0 or starting from 1 instead of 4. Others may stop at 96 and forget 100, or incorrectly count the terms in the progression. It is always helpful to check the first and last multiples of 4 within the range and then compute the count using the arithmetic progression formula to avoid off by one errors.

Final Answer:
The probability that the selected number is divisible by 4 is 1/4.