Binomial Distribution vs. Hypergeometric Distribution

Attribute	Binomial Distribution	Hypergeometric Distribution
Definition	Probability distribution of the number of successes in a fixed number of independent Bernoulli trials	Probability distribution of the number of successes in a fixed number of draws without replacement from a finite population
Number of Trials	Fixed	Fixed
Sampling Method	With replacement	Without replacement
Probability Function	P(X=k) = C(n,k) * p^k * (1-p)^(n-k)	P(X=k) = C(K,k) * C(N-K, n-k) / C(N, n)
Mean	np	n * (K/N)

Definition

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success on each trial (p). The hypergeometric distribution, on the other hand, describes the number of successes in a sample drawn without replacement from a finite population. It is characterized by three parameters: the population size (N), the number of successes in the population (K), and the sample size (n).

Formula

The probability mass function of the binomial distribution is given by the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success on each trial. The probability mass function of the hypergeometric distribution is given by the formula P(X = k) = (K choose k) * ((N-K) choose (n-k)) / (N choose n), where N is the population size, K is the number of successes in the population, n is the sample size, and k is the number of successes in the sample.

Assumptions

The binomial distribution assumes that the trials are independent and have the same probability of success. This makes it suitable for situations like coin flips or repeated experiments with the same conditions. The hypergeometric distribution, on the other hand, assumes that the sample is drawn without replacement, meaning that the outcomes are dependent on each other. This makes it suitable for situations like drawing cards from a deck or sampling without replacement from a population.

Use Cases

The binomial distribution is commonly used in scenarios where there are only two possible outcomes (success or failure) and the trials are independent. Examples include flipping a coin, rolling a die, or testing the effectiveness of a new drug. The hypergeometric distribution, on the other hand, is used in scenarios where the outcomes are not independent, such as drawing cards from a deck or sampling without replacement from a population. It is also used in quality control and auditing processes.

Mean and Variance

The mean of the binomial distribution is given by E(X) = np, and the variance is given by Var(X) = np(1-p). In contrast, the mean of the hypergeometric distribution is given by E(X) = n(K/N), and the variance is given by Var(X) = n(K/N)((N-K)/N)((N-n)/(N-1)). The variance of the hypergeometric distribution is affected by the population size and the number of successes in the population.

Sampling

One key difference between the binomial and hypergeometric distributions is how they handle sampling. In the binomial distribution, each trial is independent, and the probability of success remains constant across all trials. This makes it suitable for scenarios where sampling is done with replacement. In the hypergeometric distribution, sampling is done without replacement, meaning that the outcomes are dependent on each other. This makes it suitable for scenarios where sampling is done from a finite population without replacement.

Conclusion

In conclusion, the binomial distribution and hypergeometric distribution are both important probability distributions that are used in different scenarios. The binomial distribution is suitable for situations where there are a fixed number of independent trials with the same probability of success, while the hypergeometric distribution is suitable for situations where sampling is done without replacement from a finite population. Understanding the differences between these two distributions can help in choosing the appropriate distribution for a given scenario.