Hypergeometric Distribution

Hypergeometric Distribution

Definition

  • The hypergeometric distribution is a probability distribution that describes the probability of drawing successes in a sample drawn without replacement from a finite population.

Key Concepts and Terminology

  • Probability (P): The likelihood of obtaining a specific number of successes (k) in a specific sampling context.

  • K: The number of successes in the population. In the given example, K equals 4, as there are 4 aces in a standard deck of cards.

  • X: The number of successes in the sample drawn. In this case, X equals 2, as we are interested in the scenario where exactly 2 aces are drawn.

  • N (Capital N): The size of the total population. Here, N is 52, corresponding to the total number of cards in the deck.

  • n (lowercase n): The number of draws or samples taken from the population. This is set at 2, as we are drawing 2 cards without replacement.

  • Combinations: The number of ways to choose 'x' successes from 'k' possible successes is denoted as (\binom{k}{x}), known as "k choose x".

Example Scenario

  • Situation: Sampling cards from a deck

  • Process: Randomly sample one card from a full deck of 52 cards. After this, the card is not returned to the deck, and a second card is drawn. Then, a third card is drawn without replacing the previous cards.

  • Goal: To find the probability of obtaining exactly 2 aces from this sampling process.

Parameters in the Example

  1. Number of Aces in Population (K): 4 (The aces in a deck)

  2. Desired Aces in Sample (X): 2 (The target number of aces drawn)

  3. Population Size (N): 52 (Total cards)

  4. Number of Draws (n): 2 (Total cards drawn without replacement)

Calculation

  • The hypergeometric probability formula to compute P (the probability of obtaining x successes in n draws) is as follows:

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

  • In this case:

    • \binom{K}{X} = \binom{4}{2} (ways to choose 2 aces from the 4 available)

    • \binom{N-K}{n-X} = \binom{48}{0} (ways to choose 0 non-aces from the remaining 48 cards)

    • \binom{N}{n} = \binom{52}{2} (total ways to choose 2 cards from 52)

  • After substituting the combinations into the formula, the computed probability (P) of drawing exactly 2 aces is equal to 0.013.

Additional Information

Mean and Standard Deviation

  • Mean ($\mu$): The expected value of successes in a hypergeometric distribution.

  • Standard Deviation ($\sigma$): A measure of the amount of variation or dispersion from the mean in a hypergeometric distribution.

  • The exact formulas for mean and standard deviation of the hypergeometric distribution can be derived for specific parameters but are not provided in this context.