1. What is the Hypergeometric Expected Value?

The hypergeometric distribution describes the probability of k successes in n draws without replacement from a finite population. The expected value E[X] gives the average number of successes you would expect in a sample.

2. How Does the Calculator Work?

The calculator uses the hypergeometric expected value formula:

Where:

• \( n \) — Sample size (number of draws)
• \( K \) — Number of successes in the population
• \( N \) — Total population size

Explanation: The formula calculates the mean of the hypergeometric distribution, representing the average number of successes in a sample drawn without replacement.

3. Importance of Hypergeometric Distribution

Details: The hypergeometric distribution is crucial in scenarios involving finite populations without replacement, such as quality control, lottery probabilities, and biological sampling.

4. Using the Calculator

Tips: Enter the sample size (n), number of successes in population (K), and total population size (N). All values must be positive integers with n ≤ N and K ≤ N.

5. Frequently Asked Questions (FAQ)

Q1: When should I use hypergeometric vs binomial distribution?
A: Use hypergeometric when sampling without replacement from a finite population. Use binomial for sampling with replacement or from an infinite population.

Q2: What does the expected value represent?
A: It represents the average number of successes you would expect to get if you repeated the sampling process many times.

Q3: Can the expected value be a decimal?
A: Yes, while actual successes must be integers, the expected value can be fractional as it represents an average.

Q4: What are common applications of this calculation?
A: Quality control (defective items), ecological studies (species counts), and any scenario involving finite population sampling.

Q5: How does sample size affect the expected value?
A: The expected value increases linearly with sample size (n), as shown in the formula E[X] = nK/N.