Question

If an integer between 1 and 10,000 is randomly chosen according to the uniform distribution, approximately what is the probability of it being prime?

Short answer

The probability of a random number between 1 and 10,000 being a prime number is approximately 0.1229 or 12.29%

Step-by-step solution

Understanding Prime Numbers

A prime number is a number that has only two distinct positive divisors: 1 and itself. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, ...

Counting Prime Numbers

To find the total number of prime numbers between 1 and 10,000, a sieve algorithm like the Sieve of Eratosthenes could be used. For ease, a prime number table or a computer algorithm can be used to count the primes and it turns out there are 1,229 prime numbers between 1 and 10,000.

Calculating the Probability

The probability of an event is calculated by dividing the number of favorable outcomes (in this case the number of prime numbers) by the total number of possible outcomes (in this case the numbers from 1 to 10,000). So the probability \( P \) is given by \( P = \frac{Number\: of\: Prime\: Numbers}{Total\: Numbers} =\frac{1229}{10000} = 0.1229 \)

Key concepts

Prime Number

Understanding what a prime number is, is essential for many areas of mathematics, including probability. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it can only be divided evenly by 1 and the number itself without leaving a remainder. The first few prime numbers are 2, 3, 5, 7, 11, and so on. Note that 2 is the only even prime number, and all other even numbers can be divided by 2, hence are not prime. Recognizing prime numbers is a basic skill in number theory and essential for tackling problems that involve divisibility and factors.

For example, if you're given a random integer, say 29, and you want to determine if it's prime, you would check if any number from 2 to 28 can divide 29 without a remainder. After checking, you'd see that no number in that range can, which means 29 is a prime number. The concept of prime numbers becomes handy in cryptography, computer science, and, as we will see later, in calculating probabilities.

Uniform Distribution

When we deal with probabilities, we often assume that every outcome has an equal chance of occurring; this is known as a uniform distribution. In a uniform distribution, every event is equally likely to happen, and there is no bias towards any particular outcome. This makes it simpler when doing probability calculations as every outcome is given the same weight.

In the context of our problem, if an integer is chosen randomly between 1 to 10,000, the assumption is that every integer has the same chance of being selected. This is a key concept because it provides a baseline distribution whereby the probability of each event is the same, thus making the calculation straightforward. For instance, the chance of picking a 3 or a 5037 or any number in that range is exactly the same under this model.

Sieve of Eratosthenes

The Sieve of Eratosthenes is an ancient algorithm used to find all primes up to any given limit. It does this by iteratively marking the multiples of each prime number starting from the first prime number, 2. The multiples are marked as non-prime, and the next unmarked number is declared prime. This process continues until all the prime numbers in the specified range are identified.

The algorithm efficiently rules out the non-prime numbers, leaving only the primes on the list. When applied to the set of numbers from 1 to 10,000, as in our exercise, the Sieve of Eratosthenes can be used to count all the prime numbers in this set. It is a practical and straightforward method to identify prime numbers, especially useful when dealing with large sets of numbers.

Probability Calculation

The probability calculation is a fundamental aspect of statistics and mathematics that deals with measuring the likelihood of an event occurring. Probability is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. To calculate the probability of an event, you would divide the number of favorable outcomes by the total number of possible outcomes.

In our exercise, the event of interest is choosing a prime number at random from the integers between 1 and 10,000. To find the probability of this event, you divide the number of prime numbers (favorable outcomes) within this range by the total number of integers in the range (possible outcomes). With 1,229 prime numbers between 1 and 10,000, the probability is \( P = \frac{1229}{10000} = 0.1229 \). This translates to about a 12.29% chance that a randomly chosen number from this set will be prime.