The total number of positive integers less than or equal to 30 is 30. Therefore, the probability that a randomly selected integer is prime is:

["Understanding the Probability That a Randomly Selected Positive Integer ≤ 30 Is Prime", "When selecting a positive integer at random from the set of integers 1 through 30, a simple mathematical principle helps us determine the likelihood that the chosen number is prime. The answer starts with a clear fact: there are exactly 30 positive integers in this range. But how does that lead us to the probability of selecting a prime number?", "---", "### How Many Integers Between 1 and 30 Are Prime?", "To find the probability, we first count how many of these integers are prime numbers. A prime number is defined as a natural number greater than 1 whose only positive divisors are 1 and itself.", "Listing all prime numbers ≤ 30:", "2, 3, 5, 7, 11, 13, 17, 19, 23, 29", "We count these:\nThere are 10 prime numbers between 1 and 30.", "---", "### Calculating the Probability", "Probability is calculated as:", "[\n\ ext{Probability} = \frac{\ ext{Number of favorable outcomes}}{\ ext{Total number of possible outcomes}}\n]", "In this case:", "- Number of favorable outcomes (prime numbers): 10\n- Total number of possible outcomes (integers from 1 to 30): 30", "So,", "[\n\ ext{Probability} = \frac{10}{30} = \frac{1}{3}\n]", "---", "### Conclusion", "Thus, the probability that a randomly selected positive integer less than or equal to 30 is prime is 10/30, which simplifies to 1/3 or approximately 33.33%.", "Understanding this basic probability reinforces key statistical and number theory concepts — especially the distribution of prime numbers within a limited range. Whether for math learning, educational purposes, or real-world statistics, recognizing how small sets behave helps build intuition for larger numbers and theoretical probability modeling.", "---", "Key takeaway: Among the first 30 positive integers, 10 are prime — making the likelihood of picking a prime number 1 in 3 or about 33.3%."]