Understanding the Probability of a Specific Mean in a Random Set

In this article, we will explore the theoretical and practical aspects of determining the probability that the average or mean of a random set of 6 numbers, which are restricted between 0 and 613, will be 613. This will help us understand the nuances of probability within the realm of random sampling.

Theoretical Probability

The exact probability of the average or mean of a random set of 6 numbers equalling 613 can be determined using basic principles of probability and combinatorics. If all six numbers must be 613 to achieve a mean of 613, then we calculate the probability as 1 in 614^6. This is because each number in the set can take any of the 614 possible values (from 0 to 613), and all of them must be 613 for the mean to be 613. The formula for this is:

$P frac{1}{614^6} approx 0.00000000000000001866$

Considerations for Random Sampling

When dealing with random sampling, several factors need to be considered, including whether the numbers are integers, whether sampling is with or without replacement, and whether the numbers are uniformly distributed.

Continuously Distributed Numbers

If the selected numbers can be continuous real numbers (e.g., uniformly distributed between 0 and 613), the theoretical probability is strictly 0. In a practical sense, due to the finite precision of computer systems, the probability is vanishingly small but not exactly 0.

Without Replacement

If sampling is done without replacement, the probability is also strictly 0 because the number 613 can only be drawn once, and yet we need it to be drawn six times.

Integers with Replacement

If the numbers are integers and sampling is done with replacement, the probability is calculated as 1 in 614^6. This aligns with Paul Hudson's correct answer, as quoted in the initial problem statement: $frac{1}{614^6}$.

Multivariate Scenario: Counting the Number of Solutions

For a more complex scenario, if we want to count the number of ways six non-zero positive numbers add up to 3678, we can use combinatorial techniques like the Stars and Bars method. However, this approach would be highly mathematical and requires advanced combinatorial techniques, making it less practical for a direct calculation of probability.

Conclusion

Given the constraints, the probability of the average or mean of a random set of 6 numbers between 0 and 613 being 613 is extremely low. Specifically, it is $frac{1}{614^6}$, which is effectively zero in a practical sense.

Understanding these probabilities helps in various fields such as statistics, data analysis, and machine learning, where dealing with random sets and their properties is crucial.