10.4 Hypergeometric distribution

The hypergeometric distribution models sampling without replacement from a finite population. It's crucial for scenarios like quality control or card games, where each draw changes the probability of success.

Unlike the binomial distribution, hypergeometric accounts for changing probabilities as items are removed. It's ideal for small populations or when the sample is a significant fraction of the whole.

Hypergeometric Distribution

Hypergeometric experiment and assumptions

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• Involves drawing a sample of size $[n](https://www.fiveableKeyTerm:n)$ from a finite population of size $N$ without replacement (cards from a deck, defective items from a batch)
  • Population contains $M$ items with a specific characteristic (successes) and $N-M$ items without that characteristic (failures)
  • Sample size $n$ is fixed and known in advance
• Key assumptions:
  • Population size $N$ is finite and known (number of cards in a deck, total items in a batch)
  • Number of successes $M$ in the population is known (number of aces in a deck, defective items in a batch)
  • Each item is either a success or a failure (card is an ace or not, item is defective or not)
  • Sample is drawn without replacement, each item can only be selected once
  • Probability of success changes after each draw as the composition of the population changes (probability of drawing an ace decreases after each ace is drawn)

Hypergeometric vs binomial distributions

• Hypergeometric distribution used when sampling is done without replacement, binomial distribution assumes sampling with replacement
  • In hypergeometric, probability of success changes after each draw
  • In binomial, probability of success remains constant for each trial (flipping a coin, rolling a die)
• Population size is finite and known in hypergeometric, binomial assumes infinite population or sampling with replacement
• Number of successes in population is known in hypergeometric, not a requirement for binomial
• Hypergeometric more appropriate for smaller populations or when sample size is a significant fraction of the population (drawing 10 cards from a 52-card deck)

Hypergeometric probability calculations

• Probability mass function (PMF) of hypergeometric distribution: $P(X=[k](https://www.fiveableKeyTerm:k)) = \frac{\binom{M}{k} \binom{N-M}{n-k}}{\binom{N}{n}}$

  where:

  • $X$ = random variable representing number of successes in the sample
  • $k$ = number of successes in the sample
  • $M$ = number of successes in the population
  • $N$ = population size
  • $n$ = sample size

• To compute probabilities:

  1. Identify values of $N$, $M$, $n$, and $k$ from given problem (52 cards in deck, 4 aces, drawing 5 cards, probability of 2 aces)
  2. Substitute values into PMF formula
  3. Calculate binomial coefficients and perform arithmetic operations
  4. Result is probability of observing exactly $k$ successes in the sample

Applications of hypergeometric distribution

• Real-world problems where hypergeometric distribution can be applied:
  • Quality control: Inspecting sample of items from production batch to determine number of defective items
  • Auditing: Selecting sample of financial transactions to check for errors or fraud
  • Genetics: Studying inheritance of specific traits in a population (pea plants, eye color)
  • Surveys: Analyzing characteristics of a sample drawn from a finite population
• Steps to solve real-world problems:
  1. Identify population size ($N$), number of successes in population ($M$), and sample size ($n$)
  2. Determine number of successes ($k$) in sample to calculate probability for
  3. Use hypergeometric distribution formula to compute probability of observing $k$ successes in sample
  4. Interpret results in context of the problem

Appropriate use of hypergeometric distribution

• Hypergeometric distribution appropriate when:
  • Population is finite and known (cards in a deck, items in a batch)
  • Sample is drawn without replacement
  • Number of successes in population is known
  • Probability of success changes after each draw
  • Sample size is a significant fraction of the population (more than 5% of population)
• May not be appropriate when:
  • Population is very large compared to sample size, binomial distribution can be used as an approximation
  • Sampling is done with replacement or population is assumed to be infinite, binomial distribution is more suitable
  • Number of successes in population is unknown, other distributions or methods may be more appropriate (normal distribution for large samples)