Question

Suppose we randomly choose 100 numbers between 1 and 10,000 according to the uniform distribution. Approximately, what is the probability of all of them not being prime?

Short answer

The approximate probability of randomly choosing 100 numbers between 1 and 10,000 and all of them not being prime is very small, close to 0.

Step-by-step solution

Determine the Total Numbers and Prime Numbers between 1 and 10,000

There are exactly 10,000 numbers between 1 and 10,000. The number of prime numbers in this range is approximately 1,229. Therefore, the number of non-prime numbers is 10,000 - 1,229 = 8,771.

Calculate the Probability of Choosing a Non-prime Number

The probability is calculated by dividing the number of non-prime numbers by the total number of possibilities. That is, \(P(\text{non-prime}) = \frac{8771}{10000} = 0.8771\).

Determine the Probability of all 100 Numbers Not Being Prime

Since the choices are independent, the probability of all 100 numbers not being prime is \(P(\text{all non-prime}) = (0.8771)^{100}\).

Approximate the Result

The result is very small and is more easily understood if represented as an exponent of 10. Using a calculator or software, \(P(\text{all non-prime})\) can be approximated.

Key concepts

Uniform Distribution

When solving probability problems, it's crucial to understand the concept of uniform distribution. In uniform distribution, every outcome in a sample space is equally likely to occur. This property is fundamental in assessing the likelihood of random events within a given range. For instance, if we consider the selection of random numbers between 1 and 10,000, a uniform distribution implies that each number has an equal chance of being chosen.

One might imagine a giant jar filled with balls numbered from 1 to 10,000, and each ball has an equal probability of being drawn. In our exercise scenario, since numbers are uniformly distributed, every single number in that range is just as likely to be picked as any other. This forms the foundation for calculating probabilities in scenarios where outcomes are not biased or weighted toward any particular set of numbers. Understanding uniform distribution is vital for interpreting independent probability events and prime number calculation.

Prime Number Calculation

Prime numbers are the building blocks of the numerical system, essentially numbers that can only be divided evenly by 1 and themselves. Prime number calculation involves determining whether a number is prime and identifying all prime numbers within a specific range. This process is pivotal in our exercise since we first need to count the total number of primes between 1 and 10,000.

The approximate count of prime numbers up to a certain number 'n' can be found using the Prime Number Theorem, which states that the prime numbers become less frequent as they become larger. For our exercise, the number of primes is approximately 1,229 within the 1 to 10,000 range. By subtracting this from the total count of 10,000, we obtain the number of non-prime numbers.

Practical Implication in Problem-Solving

Recognizing how to quickly approximate the count of prime numbers streamlines the process of determining the probability of non-prime number selection. It can significantly improve the efficiency of finding solutions where prime number distribution plays a crucial role.

Independent Probability Events

The concept of independent probability events is fundamental when dealing with multiple random events, such as selecting numbers multiple times from a given range. Two events are independent if the occurrence of one does not affect the occurrence of the other. This concept is integral to our exercise in determining the probability that all 100 numbers randomly chosen are not prime.

Following the principle of independent events, the probability of each number not being prime is independent of the others. Hence, we calculate the probability of one number not being prime and then raise it to the power of 100 to represent the successive and independent choices.

• Independence Simplifies Calculations:
Leveraging the property of independent events allows us to use simple exponential calculations, making it easier to ascertain the cumulative probability of multiple non-prime selections.

Understanding independent probability in the context of prime and non-prime number selection is imperative for accurately determining the overall probability of an outcome in scenarios involving multiple, random, and independent selections.