Let x be the number of successes observed in a sample of n=5...

Let $x$ be the number of successes observed in a sample of $n=5$ items selected from a population of $N=10 .$ Suppose that of the $N=10$ items, $M=6$ are considered "successes." Find the probabilities in Exercises $11-13 .$
The probability of observing at least two successes.

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Key Concepts

Hypergeometric Distribution

The hypergeometric distribution is used to model the number of successes in draws from a finite population without replacement. It is characterized by a total population size, a set number of successes in that population, and the number of draws made. The probability mass function is given by P(X = k) = [C(M, k) * C(N – M, n – k)] / C(N, n), which calculates the likelihood of drawing exactly k successes.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways selections can be made, which is crucial for calculating probabilities in scenarios where order doesn’t matter, such as in the hypergeometric distribution. The combination function, often written as 'C(n, k)' or 'n choose k', determines the number of ways to choose k items from a total of n items, and is integral to constructing the probability formula.

Using the Complement Rule

When calculating the probability of an event that covers a range of outcomes, such as 'at least two successes', the complement rule can simplify the process. Instead of summing the probabilities for all outcomes meeting the criterion, one can subtract the sum of probabilities of the complement (fewer than two successes) from one. This method is efficient when the direct computation involves summing multiple terms.

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