In the description of a hypergeometric distribution X, given...

In the description of a hypergeometric distribution $X$, given just before Exercise 11, we see that the sample of size $n$ is taken without replacement. If the sample were taken with replacement, $X$ would be binomial with parameters $n$ and $p=r / N$. If $n$ is small relative to $N$, there is not much difference between the two methods of sampling, so a hypergeometric random variable can be approximated by a binomial random variable with parameters $n$ and $p=r / N$. (The usual rule of thumb is that $n$ should not exceed 5 percent of $N$.) Let $X$ be a hypergeometric random variable with parameters $n=5, N=500$, and $r=20$. Calculate the probability that $X=2$ and the binomial approximation to this probability.

Key Concepts

Sampling Methods: With Vs. Without Replacement

Sampling methods are differentiated by whether an item once selected is returned to the population. Sampling without replacement, as used in the hypergeometric distribution, leads to dependent draws because the composition of the population changes, whereas sampling with replacement, as in the binomial distribution, ensures each trial is independent with unchanging probabilities.

Distribution Approximation

Approximating one distribution with another involves using a simpler or more tractable model to estimate probabilities of a more complex situation. In the context of the hypergeometric and binomial distributions, when the sample size is a small proportion of the total population, the hypergeometric distribution can be approximated by the binomial distribution since the dependency between draws becomes negligible.

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population without replacement. It captures scenarios where each draw affects the subsequent probability, reflecting the dependency introduced by sampling without replacement from a limited set of items.

Binomial Distribution

The binomial distribution describes the probability of obtaining a fixed number of successes in a specified number of independent trials with a constant probability of success on each trial. This distribution applies to situations where the outcome of each trial is independent of the others, typically when sampling with replacement or when the population is large enough that removals do not significantly affect the probability.

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