Sampling Without Replacement

Unlike the binomial distribution, where trials are independent, the hypergeometric distribution describes scenarios where objects are drawn without replacement. Suppose an urn contains $N$ items of which $K$ are labeled as successes. Drawing $n$ items without replacement leads to dependent outcomes because removing one item affects the probabilities of future draws.

Calculating the PMF

The probability of obtaining exactly $k$ successes in your sample equals

$P\left(k\right)=\frac{\text{C(K,k) C(N-K,n-k)}}{\text{C(N,n)}}$

This expression counts the number of favorable combinations over the total number of ways to draw $n$ items from $N$.

Cumulative Probability

The CDF sums the probabilities for all values up to $k$. It tells you how likely it is to see at most $k$ successes in your sample. This is useful for quality inspection, card games, and any situation involving finite populations.

Start by entering the total number of items $N$, the count of successes in the population $K$, the sample size $n$, and the observed number of successes $k$. Press Calculate to display both the probability of exactly $k$ successes (the PMF) and the cumulative probability of observing up to that many.

Imagine inspecting a batch of 100 gadgets where 10 are known defects. If you randomly test 15 gadgets, the calculator can tell you the chance of finding, say, 3 or fewer faulty units. Such insight helps you set sampling plans without checking every item.

Unlike binomial trials, removing items shifts the odds with each draw. That’s why the hypergeometric model closely reflects reality when sampling from small populations. The math relies on combinations—$\text{C}$ notation—to count the number of ways each outcome can occur.

Beyond manufacturing, this distribution models card draws, ecology surveys, or any experiment where the population is finite and each selection changes the pool. By adjusting the inputs you can quickly see how different sample sizes alter the risk of missing rare items.

Understanding the Parameters

Four symbols define the hypergeometric setting and appear throughout textbooks. The population size $N$ captures how many objects exist in total; this might be cards in a deck, balls in an urn, or widgets on an assembly line. Out of those $N$ objects, $K$ are labeled “successes” while the remaining $N-$ K$$ are considered failures. When we draw $n$ objects without replacement, we observe some number $k$ of successes. Although the letters are short, it helps to think of them as entire stories: the narrative of a lottery sample, a quality-control batch, or a genetic cross.

Step-by-Step Manual Calculation

Performing a hypergeometric calculation by hand highlights the mechanics hidden behind the calculator’s instant answer. The process unfolds in three conceptual stages. First, count how many ways the desired outcome can occur: choosing $k$ successes from the $K$ available and $n-k$ failures from the $N-$ K$$ remaining. Second, count how many overall samples are possible by selecting $n$ from $N$. Third, divide the favorable count by the total count. Every symbol in the formula expresses one of these pieces. Writing out each step keeps you honest and reveals where mistakes might creep in.

Worked Example in Detail

Consider a card game where a standard 52‑card deck contains four aces. If you draw ten cards at random, what is the probability that exactly two are aces? In this case $\mathrm{N=52}$, $\mathrm{K=4}$, $\mathrm{n=10}$, and $\mathrm{k=2}$. The calculator first evaluates $\text{C(4,2)}$, the number of ways to pick two aces. It then multiplies that by $\text{C(48,8)}$ because eight non‑aces must accompany them. Finally it divides by $\text{C(52,10)}$, the total ways to draw ten cards. The resulting probability, about 0.0399, reflects how rare it is to see exactly two aces in such a draw. Running the same scenario with different values of $k$ reveals the full distribution, letting you gauge the odds of one ace, three aces, or any other outcome.

Mean and Variance

The hypergeometric distribution has a built‑in expected value and spread just like more familiar distributions. The mean, or expected number of successes, equals $\mathrm{nK/N}$. Intuitively, sampling ten cards from a deck with four aces yields an average of $104/52$ ≈ 0.769 aces per draw, even though any particular draw may produce none. The variance quantifies how widely the outcomes are spread and is given by , often rewritten as $n\frac{K}{N}\frac{\mathrm{N-K}}{N}\frac{\mathrm{N-n}}{\mathrm{N-1}}$. This factor $\frac{\mathrm{N-n}}{\mathrm{N-1}}$ is the finite population correction; it shrinks the variance as the sample approaches the population in size because there is less uncertainty when you are drawing most of the items.

Comparing to the Binomial Distribution

Students often encounter the binomial distribution first and wonder when the hypergeometric is necessary. The binomial assumes independent trials with replacement or a vast population where individual selections barely affect the odds. When the sample size is small relative to the population, the binomial is a good approximation and easier to compute. However, as $n$ becomes a substantial fraction of $N$, ignoring the changing probabilities introduces noticeable error. A helpful rule of thumb is that if the sample is more than about five percent of the population, the hypergeometric model provides a more faithful picture.

Real-World Applications

Hypergeometric reasoning appears whenever resources are limited and selections are made without replenishment. Quality engineers inspect batches of electronics to estimate defect rates before a product ships. Ecologists capture and tag animals, then recapture a sample to infer population sizes. In card games such as poker or collectible deck-building, calculating draw probabilities helps players determine the risk of going for a particular strategy. Even genetics uses hypergeometric thinking: in Mendelian inheritance problems, gametes combine without replacement, producing distributions of traits in offspring.

Interpreting Calculator Output

The PMF value tells you the likelihood of a single outcome. The CDF aggregates probabilities up to $k$, revealing the chance that the count of successes does not exceed that threshold. The calculator now also reports the distribution’s mean and variance, giving you a sense of typical results and variability. When comparing sampling plans, a small variance indicates results that cluster tightly around the mean, while a large variance signals a wide spread of possible outcomes.

Common Mistakes and Edge Cases

Several pitfalls can lead to incorrect conclusions. Forgetting that $k$ cannot exceed either $n$ or $K$ is a frequent oversight; the calculator checks these conditions and alerts you if they are violated. Another error is using the hypergeometric model when sampling with replacement, in which case the binomial distribution is more appropriate. Finally, watch out for extreme inputs like $\mathrm{n=N}$; this degenerate case guarantees all successes or all failures depending on $K$.

Using the Calculator Effectively

To get the most out of the tool, start with a scenario and adjust one parameter at a time. Observe how increasing the sample size raises both the mean and the variance. Try doubling $K$ to see how a richer population of successes changes the distribution’s shape. If the numbers become large, remember that the calculator uses efficient algorithms to compute combinations without overflowing, so you can safely explore populations in the thousands.

Summary

The hypergeometric distribution captures the nuances of sampling without replacement. By understanding its parameters, formula, and practical context, you can model real-life selection processes with confidence. The expanded calculator on this page not only finds the probability of specific outcomes and cumulative totals but also reports the mean and variance while guarding against invalid input. Use it to design inspection routines, study card odds, or analyze biological experiments—anywhere the act of drawing changes the composition of what remains.