Finite Sampling Probability Calculator

When sampling from a finite population without replacement, the probability of obtaining a specific number of "special" items follows a distinct mathematical pattern known as the finite sampling distribution. This distribution is fundamentally different from binomial sampling because once an item is drawn, it cannot be selected again, making each subsequent draw dependent on previous outcomes.

Consider a population containing N total items, where K of them possess a specific characteristic (let's call them "successes"). If we randomly draw n items from this population without replacement, the probability of obtaining exactly k successes follows a precise combinatorial formula.

Mathematical Foundation:

The probability is computed using the ratio of favorable outcomes to total possible outcomes:

$$P(X = k) = \frac{\binom{K}{k} \cdot \binom{N-K}{n-k}}{\binom{N}{n}}$$

Where:

• $\binom{K}{k}$ represents the number of ways to choose k successes from K available successes
• $\binom{N-K}{n-k}$ represents the number of ways to choose the remaining n-k items from the N-K non-success items
• $\binom{N}{n}$ represents the total number of ways to choose any n items from the entire population

Validity Constraints:

For a valid probability calculation, k must fall within feasible bounds:

• k ≥ max(0, n - (N - K)): We must draw at least enough successes to fill the sample if there aren't enough non-successes
• k ≤ min(n, K): We cannot draw more successes than either exist in the population or than we're drawing total

If k falls outside these bounds, the probability is 0.0 (an impossible event).

Your Task:

Write a function that computes the probability of obtaining exactly k successes when drawing n items without replacement from a population of size N containing K success items. Return the result rounded to 4 decimal places.