Understanding the Probability of Prime Numbers in a Given Range

When discussing divisibility within a specific range, understanding the basic definitions and properties of numbers is crucial. Specifically, the question of finding the probability of a number between 1 and 100 that is divisible by both 1 and itself requires a clear understanding of divisibility and prime numbers. This article delves into these concepts, providing clarity and valuable insights.

Divisibility by 1 and the Number Itself

Every integer is divisible by 1, and every integer is divisible by itself. Hence, any number between 1 and 100 is divisible by both 1 and itself. This means the total number of integers in the range from 1 to 100 is 100, and every single one of these integers will satisfy this condition.

Probability Calculation

The probability can be calculated using the formula:

Probability Number of favorable outcomes / Total number of outcomes

In this case:

Probability 100 / 100 1

Therefore, the probability of getting a number between 1 and 100 that is divisible by 1 and itself is 1, or 100%.

Prime Numbers and Divisibility

However, if the question is about the probability of getting a number between 1 and 100 that is divisible by 1 and itself only, the question shifts to finding prime numbers within that range. Prime numbers are integers that have exactly two distinct positive divisors: 1 and the number itself. This means that such numbers are only divisible by 1 and themselves.

Prime Numbers Between 1 and 100

To find the number of prime numbers between 1 and 100, we need to recall the prime numbers in this range. These are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Counting these, we find there are 25 prime numbers between 1 and 100.

Probability of Getting a Prime Number

Now, we can calculate the probability of choosing a prime number from the numbers between 1 and 100:

Number of prime numbers between 1 and 100 25

Total numbers between 1 and 100 100

Probability of getting a prime number 25 / 100 1/4 0.25 or 25%

Therefore, the probability of getting a number between 1 and 100 that is divisible by 1 and itself only is 25%.

Conclusion

When the question asks for a number between 1 and 100 that is divisible by 1 and itself, the straightforward answer is that such a number must be a prime number. Based on the definition of prime numbers and the count of such numbers within the specified range, the probability of picking a prime number is 25%, or 0.25.

Understanding the nuances of divisibility and prime numbers is key to solving such problems accurately. Whether you are approaching from a basic divisibility perspective or delving into the realms of prime numbers, the concept remains fundamental in number theory.