Hypergeometric Distribution

Lesson Goals

• Calculate probabilities using the hypergeometric distribution.

Hypergeometric Distribution

• Defined by:

  • A specified number of trials (n)

  • A specified number of successes (x)

• Distinction: Deals with dependent trials rather than independent trials.

  • Dependent events: One event occurring affects the probability of another.

  • Example: Choosing cars without replacement.

Properties of Hypergeometric Distribution

1. Trials Selection:

   • Each trial involves selecting one item from a population of size N.

   • Results in success or failure (known population).

2. Number of Trials:

   • The experiment consists of n trials.

3. Possible Successes:

   • Total possible successes in the population is k.

4. Dependent Trials:

   • Trials are dependent on each other.

5. Random Variable:

   • Hypergeometric random variable (X) counts the number of successes in n trials.

6. Mean and Variance:

   • Mean (μ): μ = n * (k/N)

   • Variance (σ²): σ² = (n * k * (N - k) * (N - n)) / (N² * (N - 1))

Probability for Hypergeometric Distribution

• For a hypergeometric random variable (X), the probability of obtaining x successes in n trials is:

  • P(X = x) = ( \frac{\binom{k}{x} \cdot \binom{N - k}{n - x}}{\binom{N}{n}} )

  • Where:

    • N = total items in population

    • n = number of trials

    • k = number of successes in population

    • x = number of successes in trials

• Rounding Rule:

  • Round probabilities to four decimal places.

Comparison of Discrete Probability Distributions

• Table Summary: Hypergeometric, Binomial, Poisson distributions.

• Characteristics:

  • All are discrete distributions.

  • Independent Trials:

    • Binomial & Poisson: Independent

    • Hypergeometric: Dependent

  • Fixed Number of Trials:

    • Binomial & Hypergeometric: Fixed

    • Poisson: Not fixed

Conclusion

• This lesson provided a comprehensive overview of the hypergeometric distribution, highlighting its unique properties and its calculation.