Hypergeometric distribution

Definition

The hypergeometric distribution is a probability distribution that describes the likelihood of a certain number of successes in a sequence of draws without replacement from a finite population. This distribution is particularly useful when the total number of items in the population, the number of successes in that population, and the number of draws are known, allowing for the calculation of probabilities based on specific scenarios in sampling.

5 Must Know Facts For Your Next Test

1. The hypergeometric distribution applies when sampling from a finite population without replacement, meaning once an item is drawn, it cannot be drawn again.
2. It is defined by three parameters: the total number of items in the population (N), the number of successful items in that population (K), and the number of draws (n).
3. The probability mass function for the hypergeometric distribution can be expressed as $$P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}}$$ where k is the number of observed successes.
4. In contrast to the binomial distribution, which assumes independence between trials, the hypergeometric distribution accounts for changes in probabilities as items are drawn from the population.
5. Common applications include quality control processes and ecological studies, where researchers may need to determine probabilities related to specific attributes in a sampled group.

Review Questions

• How does the hypergeometric distribution differ from the binomial distribution in terms of sampling methods?
  • The key difference between the hypergeometric distribution and binomial distribution lies in their sampling methods. The hypergeometric distribution is used when sampling without replacement from a finite population, which means that each draw affects subsequent probabilities. In contrast, the binomial distribution assumes that each trial is independent and allows for replacement, treating each trial as having constant success probabilities. This makes the hypergeometric model more suitable for scenarios where sample size impacts outcomes.
• Calculate the probability of drawing 3 red balls from a box containing 5 red balls and 10 blue balls when drawing 5 balls without replacement using the hypergeometric distribution.
  • To calculate this probability using the hypergeometric distribution, we define our parameters: N (total balls) = 15, K (successful red balls) = 5, n (draws) = 5, and k (observed successes) = 3. The probability can be calculated as $$P(X = 3) = \frac{{\binom{5}{3} \binom{10}{2}}}{{\binom{15}{5}}}$$. This equals $$\frac{{10 \times 45}}{{3003}} \approx 0.1498$$ or 14.98%.
• Critically evaluate how understanding the hypergeometric distribution can impact decision-making processes in fields like ecology or quality control.
  • Understanding the hypergeometric distribution is crucial in fields like ecology and quality control because it allows professionals to accurately assess probabilities related to finite populations without replacement. In ecology, for instance, it helps estimate species abundance and diversity by analyzing sampled populations. In quality control, it aids in evaluating defect rates among batches when inspecting a limited number of items. This understanding informs better decision-making by providing precise risk assessments and improving resource allocation based on likely outcomes.

Related terms

Binomial Distribution: A probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Sampling Without Replacement: A sampling method where selected individuals or items are not returned to the population, influencing the probabilities of subsequent selections.

Combinations: Mathematical expressions used to calculate the number of ways to choose a subset of items from a larger set, regardless of the order in which they are chosen.